On the stability of running of locomotivesrspa.royalsocietypublishing.org/content/royprsa/121/788/585.full.pdf · riding qualities of a design can be brought within the realm of rational

Stability o f Running o f Locomotives. 585

value of tfR, and that Messrs. Barling and Webb’s formula would give 2-6 tons as the value of the stress at which wrinkling occurs. Further comment as to the insufficiency of these formulae is unnecessary.

The author desires to express his thanks to Sir John Petavel, F.R.S., for his constant help and encouragement when some of the work was being carried out in his laboratory when he was Professor of Engineering at Manchester; to the Superintendent of the Royal Aircraft Establishment and the Air Ministry for permission to publish some of the results obtained whilst the author was at the Royal Aircraft Establishment; to his assistants at the Royal Aircraft Establishment, Mr. I. J. Gerard, M.Sc., and Miss Pettifer, B.Sc. to his present research assistant, Mr. A. J. Newport, B.Sc., for assistance in carrying out some of the experiments, and especially to his friend, Mr. R. V. Southwell, F.R.S., for many helpful discussions in problems connected with this and other investigations.

On the Stability o f Running o f Locomotives.By F. W. Carter, M.A., Sc.D., M.Inst.C.E., M.I.E.E.

(Communicated by A. E. H. Love, F.R.S.—Received July 25, 1928.)

1. Introduction.In a paper entitled “ The Electric Locomotive,” read before the Institution

of Civil Engineers in 1916,* the author gave the beginnings of a rational discussion of the riding qualities of electric locomotives, as affected by their fundamental type. I t was shown that, aside from all questions of unevenness of track or of the nature of the springing and equalizing arrangements, there might exist in a locomotive an inherent tendency to deviate from a motion of pure progression, which, although perhaps within the control of the wheel flanges and therefore deemed unobjectionable on good track, might give reason for anxiety if poor track were encountered or other external circumstances conspired with inherent tendencies to bring about a condition involving danger of derailment. I t is the intention, in the present paper, to discuss the matter further with the view of determining to what extent inherent- riding qualities of a design can be brought within the realm of rational investigation.

* ‘ Proe. Inst. C.E.,’ vol. 201, p. 221.

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586 F . W . C a r te r .

In applying mathematical analysis to such a subject as the present, it is necessary to assume that all parts fulfil their intentions ; though, actually, few do so perfectly. The track is assumed smooth, uniform and level; the clearances required by mechanical considerations, except in so far as they enter into the problem, are assumed to be zero ; lubricated slides are assumed frictionless ; and so on. I t must be postulated that a tendency shown by a machine in which the parts fulfil their intentions will, in general, evince itself in some form in such machines as can actually be constructed. The observed phenomena are likely to reflect the conditions in being less definite, less regular, and more catastrophic than the analysis would indicate ; and the results of the present analysis are not expected to possess metrical value of a high order. This, however, is in the nature of the subject and does not detract from the value of the study of tendencies, and the importance to the designer of a knowledge of conditions tending to produce stable running.

2. General.From the present point of view, a locomotive consists of one or more rigid

frames, each symmetrical about a longitudinal central plane, supported by springs from a group of axles, and carrying appropriate driving mechanism. The purpose of a frame is to constrain the axles, so that they keep in certain transverse vertical planes fixed in the frame, with the centre point of each on the longitudinal plane of symmetry, whilst permitting to the axles freedom of rotation and a certain freedom of motion in their several vertical planes. The locomotive may consist of one or more units, and each unit may comprise a single main frame, or a main frame with one or two auxiliary frames,* these being capable of prescribed movement with respect to the main frame, and biassed towards symmetrical positions by prescribed forces. Each axle bears a pair of wheels, rigidly attached to i t ; and the treads of the wheels are coned slightly, tapering outwards, (1 in 20), the rails on which they run being generally canted inwards to a like extent.

In the performance of its functions the frame imposes constraint on the axles and vice versa, giving rise to those forces at the wheel-treads which are the ultimate guiding forces of the locomotive. The consideration of these forces is the characteristic and fundamental feature of the present study.

* The arrangement of a locomotive unit is, in this country, usually symbolized by a group of three figures, representing successively the number of wheels in the fore auxiliary truck, the main truck, and the rear auxiliary truck.



Stability o f Running o f . 587

3. On the Action between Wheel and Rail.The area of contact between wheel and rail has a shape depending on the

initial form of the surfaces in contact. When the rail is new, the radius of the transverse section of its head or running surface, is usually smaller than the radius of a wheel; and the contact area is bounded by an ellipse having its major axis in the direction of the rail. The head is flattened by use however, and with a worn rail the major axis of the contact ellipse is transverse to the rail. When the radius of the section of the rail head is equal to that of the wheel, the contact area is circular, having a diameter of the order of 14 mm.*

As the wheel rolls, the area of contact travels along the rail and about the periphery of the wheel. If the force between wheel and rail is normal to the contact area, points which come into contact remain so engaged throughout the period of contact. If, however, the force has a component tangential to the contact area, the engagement or meshing of the surfaces is confined to an initial portion of the period of contact, the final portion being characterized by a relative slipping of the surfaces against the tangential force.f In this case, moreover, the engagement is not between unstrained, or equally strained, surfaces, but between surfaces which are oppositely strained, the tangential force causing extension of the surface of one member, opposed by contraction of the other, and vice versa. In consequence of this, the rolling displacement of the wheel is, under the influence of the tangential force, accompanied by a creeping displacement in the general direction of the force. The latter displacement is proportional to the former, and is otherwise a function of the tengential force.

Defining “ creepage ” as the ratio between the creeping and rolling displacements, and designating it by y, the tangential force acting at the tread of the wheel may be written^

F = - / y . (1)

The value of / has been determined by the author ( . cit.) for a certain casein which the tangential force is in the direction of rolling, its value for steel wheels and rails being about

/= 9 3 [ a M * [ l + ( ! —# ! (2)* See Love’s ‘ Elasticity,’ par. 138. The figure given is based on a pressure of 8,000 kg.,

and a radius of 600 mm.j See “ On the Action of a Locomotive Driving Wheel,” by F. W. Carter, ‘ Roy. Soc.

Proc.,’ A, vol. 112, p. 151 (1926).X The notation used is given in an Appendix, pp. 609-11.



588 F . W . C a r te r .

in which a is the radius of the wheel in mm .; l is a linear magnitude, representing the effective length of the contact area transverse to the rail in mm.; w is the total pressure between the surfaces, ., the weight borne by the

wheel, in kilograms ; q is the ratio of the tangential force to the maximum that could be sustained without skidding or bodily slipping; and / is in kilograms.* I t will be seen that/increases in the ratio 1 : 2 as F falls from its maximum value to zero.

The effective value o f/is a matter of importance in the present investigation. The formula given above is insufficient on two counts, viz., the effective value of l is unknown, and probably varies with the condition of the track, and the effective value of q is matter for conjecture. These defects are properly taken into account by reference to general experience in the subject, or to experiment devised to determine a quantity, A, such that

/ = A (awf. (3)In the absence of such empirical knowledge, we herein assume

A = 800,

which may be interpreted as corresponding to — 20 mm. and q — 0-15.Whether the creepage per unit tangential force is the same when the force

acts at right angles to the direction of rolling as when the directions coincide or oppose, is matter for conjecture. The most that can be asserted with confidence at present is that, depending as it does on the difference in initial strain of the surfaces which engage one another, it is of the same order whatever the direction of the force, and no reason is apparent why its value should not be independent of this direction. Until fuller knowledge is available, accordingly, we assume the creepage coefficient / given by equations 2 or 3, to be applicable whatever the direction of the tangential force/

We may remark that a lack of precise knowledge in these matters does not vitiate general conclusions. We have given expression to our ignorance by introducing the quantity A. If we may assume that, for the several wheel- groups of a locomotive, this quantity is sensibly a constant within the rough limits that the nature of the subject renders necessary, the absence of knowledge of its actual effective value only affects absolute results. The curves of figs. 6 to 9 and 11 to 14, for instance, are affected as to scale, but not as to shape.

* With lengths in inches and forces in pounds, the coefficient in the equation becomes 3500 instead of 93.

t In the author’s book * Railway Electric Traction ’ (Arnold), different creepage coefficients were assumed for forces along and transverse to the rails. No new phenomena were, however, indicated.




The extent to which creepage motions are involved in locomotive running will now be considered. The maximum tractive effort of a wheel before skidding is usually of the order of 35 per cent, of the pressure on the tread.* Assume a — 625 mm., I = 25 mm., w —9000 kg. and 1, equation 2 gives

/ = 1 • 1 X 106 kg. Hence the creepage is

0-35 x 9000/(1-1 X 106) = 2-86 x 10~3.

If the treads of a pair of wheels are 1 -5 m. apart (which corresponds approximately with standard gauge) the combination can be caused to traverse, without skidding, any curve of radius not less than R metres, given by

(R + 0-75/(R - 0-75) = (1 + 2-86 . KT3)/(1 - 2-86 . KT3) or

R = 262 metres.

This is a small radius for a railway curve, and locomotives therefore take normal service curves without skidding the wheels. Again, the pair of wheels could be run on straight track, without skidding, if their respective radii were 625 (1 + 2-86 X 10-3) mm. and 625 (1 — 2-86 X 10~3) mm. Since the conicity of the wheels is 1 in 20, this difference corresponds with a movement of the pair from the central position of 20 X 625 X 2-86 X 10~3 mm., i.e., 35-8 mm., or a total play of 71-6 mm., which is certainly more than double the movement which the flanges would permit.f I t may be concluded that the motions of deviation from pure progression which are possible to a locomotive are of the nature of creeping rather than slipping.

4. Simple Rolling of Wheels and Axle.Fig. 1 represents a pair of railway wheels mounted on axle, running on rails

canted X to correspond with the coning of the wheels. Taking axes of co-ordinates in the plane of the track, with the centre line of the track as x-axis, let (xy) be the co-ordinates of the centre of the axle, and ^ its inclination with the //-axis, in the plan view. The effective radius of the wheel A is

* In electric locomotives having independently driven axles, experience has shown that, under ordinary circumstances, a tractive effort of 25 to 30 per cent, of the weight on drivers can be sustained ; but the pull of the draw-bar re-distributes the pressure on the several axles, decreasing that on the forward axles, which accordingly limit the tractive effort. The limiting ratio for these is usually at least 35 per cent.

t The play permitted between flanges and rails, as fixed by the Berne Conference, May 15, 1886, is, minimum 15 mm., maximum 35 mm. See 6 Traite de Stability du Materiel des Chemins de Fer,’ par Georges Marie, page 306.

2 RVOL. CXXI.— A.



590 F . W . C a r te r .

9

Fig. 1.—Diagram of Wheels and Axle.

a -j- a y,and of the wheel B, a — Ay. The co-ordinates of the two points of contact are—

of A {x — by -f- b), of B -f b^, y —The component displacements are, therefore,

of A {dx — bdfi, dy], of B {dx + bd^, dy} ; (4)but the components of rolling displacement are,

of A {(a + Ay)d0, a^dC], and of B {( — Ay)c?0, ; (5)and, accordingly, for simple rolling,

dx = add, dy = atydft, = — Ayd0.Thus

ab = °' ^

The motion is, therefore, sinuous and repeats in distance,

X ^Z rW a b T k- (?)

5. Constrained Rolling of Wheels and Axle.It has been shown that if appropriate constraint be applied to the axle,

the rolling motion is accompanied by creeping. The components of creepage displacement are, from 4 and 5 above,

of A {dx — bd'ty — (a -j- Ay) dO, dy

of B {dx + — (a — Xy)d0The components of creepage are

of A {— ---- 1 — ^

a'\)dC)

dy — a^dC]

of B

\ad% aj adO ^ j1

x + ( ^ + x S\aa 9 ady

ad% * } Jb (9)




The component forces at the wheel-treads are, accordingly,

at A | — /

at B j—/

dx.add \a au a/J

+ x -aau \aa t)

- /

/

\dy.add

dy_add

This system of forces is equivalent to —1. A longitudinal force at the centre of the axle, of value,

1 h

2. A transverse force at the centre of the axle, of value,

Yx =

y (io)

(ii)

( 12)

3. A couple about a vertical axis, having moment

G1= - 2 f ( - ^ + -k-y). (IS)\a d d a

If the system of forces be referred to some fixed point, Q (fig. 2), on the centre line of the locomotive or truck (in the plan view), having co-ordinates

1(

---------------------------------------------- 1i-----------------------

^ c ,i (xq) S G

L - k ----- --------i----------— ---------- L_____________________

Fig. 2.—Diagram of Truck.

(xy), and at distance c behind the axle, the components of the system become, to the degree of approximation required,

X = (14)

Y = — 2/

G = - 2/ (62 + c2) d±add

dy_add

/- (1 > ; ) " * + • ! + » ; » ]

2 r 2

(15)

(16)



592 F. W . C a r te r .

Since/is large compared with the greatest possible value of X, differsbut little from unity, and dx may be substituted for add in equations 15 and 16.

6. Equations of .

Let the centre of gravity of the portion of the locomotive affected by the group of wheels under consideration be distant, h, ahead of the reference point Q (fig. 2). Let m be the mass of the part, and its moment of inertia about the vertical through the centre of gravity. Let X'Y'G' be components and moment of other forces acting on the part, such as those produced by auxiliary trucks, these being reduced to some point S on the axis of symmetry of the locomotive, distant s ahead of the reference point for the wheel-forces and motion. The equations of motion of the part are—

= X' + SX . (17)

+ / < f ) = Y' + S Y. (18)

mk2= G' + SG — (Y' + SY) + sY'. (19)

Equation 19 may be replaced by—

m (A2 + k2) f i + mh f | = G' + EG + '.v at2dt2

(20)

In these equations, the summations extend over the several axles of the group under consideration. Equation 17 has no particular interest in regard to matters now under consideration, being concerned with the normal progression of the locomotive. We do not take account of the effect of the forward acceleration of the locomotive on the phenomenon of deviation. Accordingly, if Y is the velocity of progression, it is permissible to write—

d t 2and, taking account of the remark which follows equation 16, equations 18 and 20 reduce to

mV2jlx2 ‘ dx2. + 2 S / .ax dx Y', (21)

dx2

a /




These are the equations which determine the deviation of the motion from normal progression. Since, in applying them to problems of particular locomotives, expressions of considerable length and complexity arise, it is expedient to adopt a notation which reduces the amount of writing to the minimum. The notation used is, for convenience, given in an appendix to the paper. In terms of this notation, writing D for dJdx, equations 21 and 22 become—

[aD2 + D ]y-f- [aM)2 + cD — 1] ^ = |kY ', (23)[a/?I)2 -f- cD -f- s] y -f- [a (A2 -f- k2) D2 -f- d2D + b — c] ij;

= *k (G' + (24)

7. Locomotives of One Axle-Group.For the locomotive running alone, Y' = 0, G' = 0: equations 23 and 24

therefore give—{a2PD4 + a [h2 + k2 + d2 - 2hc] D3 + [d2 - c2 + a (b - c + s D2

+ [b — sc] D + s} d; = 0. (25)

sc >

This equation indicates stable motion of progression if*

3 - ( h2 + k2 + d2 — 2^c)s — (b — c -f — s^) (b — sc)

(b - sc)2 )^ K }

The condition b — sc !> 0 implies, as shown in the Appendix, that the wheels of the several axles differ in diameter, and that the smaller wheels are ahead. This is not usual in a locomotive having a fixed wheel-base, but only on these conditions is stable running possible, and then only to the speed limit given by the inequality 26.

A numerical example may be of interest as showing the order of quantities involved. Take dimensions and weights as shown in fig. 3, standard gauge (b = 0-75 m.), the truck running with small wheels ahead.

■0-8m-

l-75m------- 1 75m

8000 kg 800 0 kg 8000 kg 6000 kg

Fig. 3.—Truck carried on rigid wheel-base.

* For conditions of stability of a motion represented by a linear differential equation having constant coefficients, see Routh’s ‘ Advanced Rigid Dynamics/ pars. 290-307.



594 F . W . C a r te r .

For the front ax le / = 1 -24 X 106, for the three rear axles/ = 1-96 X 106, hence f = 7-12 X 10°. We take the reference point at Gr, accordingly,

c = — 0-0969 s = 0-0576 b = 0-0180

b — c = 0-1149 b — sc = 0-02354

d2 = 4-281 d2 — c2 = 4-271.

k2 — 4 h = 0 m = 60,000 kg.

a = 4-296 X 10"4V2.The inequality 26 therefore gives

4-296 v 10“4 V2 ,•0-02354 > — - {0-477 — 0-0027 + 0-0003}.

We deduce that the running becomes unstable at a speed of 22-2 metres per second or about 50 miles per hour. The three terms within the bracket have been kept separate in order to show that the first alone is of importance : the second arises from the coefficient of a in the D2 term in equation 25, and the third from the first, term in this equation. We might, accordingly, without sensible error have written for equation 25

{a ( h2 + k2 + d2 — 2Ac) D3 + (d2 - c2) D2 + (b — sc) D + s} = 0. (27)

Moreover, within the limits imposed by practicability, we can readily obtain an approximate algebraic solution of this equation. Assume

4- = +0 ** = +0 e({+*’",with Z, — it\as a first approximation, where

7] = d2 - c2/(28)

Proceeding to a next approximation, we obtain

a ( h2 + k2 + d2 — 2hc) (b — sc)2 (d2 — c2)2 2 (d2 — c2)"

(29)

The condition for stability, viz., that £ should be negative, agrees with 26, with the omission of the last two terms within the brackets. We note that the motion is periodic in distance

(30)

This is, of course, independent of the reference point..*

* See Appendix.




Since the example given above does not refer to a normal type of locomotive of one axle group, we discuss briefly the more normal type shown in fig. 4.

l-25m

1 5 m -----►------ 15m

8000 kg

Fig. 4.—Locomotive carried on rigid wheel-base (0-6-0).

Here, taking the centre as reference point for the motion,

/ = 1-79 X 106 f = 5-37 X 106 48,000 kg.a = 4-56 X 10-4 V2 b — 0, c = 0,s = 0-06 d2 = 2-062 = 0, k = 1-2,

whence7j = 0-1705 l = 1-126 X n r 5v 2X = 36-8 m. = 4-15 X 1(T4Y2.

Thus, at V = 15 m.p.s., the amplitude of deviation increases by about 10 per cent, in each complete period ; at V = 20 m.p.s., the increase is about 18 per cent., whilst at Y = 25 m.p.s., it is about 30 per cent.

The locomotive carried on a single axle-group is much used in working freight trains ; but is not employed for high-speed running on account of proclivities indicated in the foregoing discussion. The hauling of a train or tender has some tendency to stabilize the running,* but the effect is not great, being of the second order. The use of side buffers (as distinguished from centre buffers) produces a tendency of the first order making for stable running, as long as they remain in compression.

The periodic digressive motion, at least in its more violent manifestation, is known as “ nosing.” I t is not, properly speaking, an oscillation, but is the result of instability of a pure motion of progression. In the ordinary locomotive of the type, the instability is to be attributed entirely to the inertia of the body, but locomotives having unequal wheels, in pairs, may, as has been shown, have stability or instability independently of the inertia, according to the location of the wheels. Nosing has in its origin nothing to do with the spring or equalizing systems, although it is likely to become noticeable, if not to be actually enhanced, at such speed that the period of the motion, in time,

* ‘ Proc. Inst. C.E.,’ vol. 201, p. 251 (1916).



596 F . W . C a r te r .

agrees with a natural period of oscillation of the locomotive. The periodic distance is, as has been shown, considerably greater than the length of a rail, and nosing is therefore unaffected by the periodicity in the track.

8. Locomotives of Two Axle-Groujps.The locomotives now to be discussed have, in general, one group of axles

associated with the chief mass of the locomotive, and this is designated the main group ; the other group constitutes an auxiliary truck, and the associated parts are usually of comparatively small mass. The auxiliary truck pivots about a point on the centre line of the main frame, the pivot being sometimes situated at a definite point in the truck itself, as in the Bissel, and sometimes permitted to traverse the truck, as in the bogie. The design of locomotives in respect of these matters has hitherto been determined principally with reference to the need for steering the locomotive on curves, and providing a flexible wheel-base over which to distribute its weight; but since it has much influence on the running qualities now under consideration, the problems should be correlated.

The auxiliary truck is, in general, biassed towards a symmetrical central position by means of springs, or equivalent devices, which apply the steering forces to the main frame. In many locomotives the centring devices are so designed that considerable force is necessary to displace the truck from the position of symmetry, although, once displaced, the variation of the force is comparatively small, or even zero. I t is not practicable to take account of this discontinuity, and the centring forces will, in all cases, be assumed proportional to the displacement from the position of symmetry.

We take the pivotal point on the main frame as reference point, both for the motion and for the forces between the two elements. We use dashed 1 etters to distinguish auxiliary from main trucks. We assume a centring force, — 2Y ( y — y'), to act on the main truck at the pivot, and an aligning couple, — 2G — d/), to act about the pivot, thus including all cases in one formula. The equations of devious motion are (see equations 23 and 24),

(aD2 + D )-y + (aAD2 + cD - 1) + kY (y — y') = 0, (31)(a'D2 + D) y’ + (a'A'D2 + c'D - 1) tj/ + k'Y (y' = 0, (32)(aM)2 + cD + s) y + [a (h2 + F) D2 + d2D + b - c] <], -f kG (^ - V) = °>

(33)

(a'A'D2 + c/D + s') y’ + [a' (h'2 + Jc'2) D2 + d'2D + b' - c'] <j/+ k 'G ( ^ - ^ ) = 0. (34)



These give, as the equation which determines stability,

aD2+ D + k Y - k Y a/*D2+ c D - l 0

—k'Y a 'D 2+ D + k 'Y 0 a T D 2+ c 'D - l

aAD2+ c D + s 0 a(A2+ ft2)D2+ d 2I) - k G+ b - c - f k G

0 a T D 2+ c 'D + s ' —k'G a '(/*'2+&'2)D2+ d '2D+ b '—c'-fk 'G

= 0. (35)

This is the characteristic equation of the running of locomotives of two axle- groups, in its general form. The expansion of the determinant is long ; but, since it is symmetrical as regards the trucks, it is not difficult.

I t is expedient to divide the terms into four categories, viz., those independent of Y and G, those involving Y only, those involving G only, and those involving both Y and G. The terms of the first category form the product of two expressions similar to equation 25. They are as follows :—

V W 2D8 + aa ' { a k2 ( h '2 + k'2 + d 2 - 2 h'c') + a + + d2 - D7

+ {a 2k2 (d'2 - c '2) + a '2F 2 (d2 - c2) + aa ' ( h2-f + d2 - + k'2 + d '2 -

-f- aa ' [aft2 (b' — c' -f — s -f- a (b — c -j — sftJjjD'1

+ {a (h2 + k2 - f d2 - 2 he)(d '2 - c'2) + a ' ( h'2 + k'2 + d '2 - 2ft'c') (d2 - c2)

-}-aa'[(ft2-|-ft2-|-d2—2ftc) (b '— c'-\-h'— s'h') -j- (ft'2-|-ft'2-}-d'2—2ft'c') (b— sft)]

+ a2F ( b ' - s 'c ') + a '2ft'2( b - sc)}D5

+ {(d2 - c2) (d'2 - c'2) + a (d'2 - c'2) (b - c + h -+ a ' (d2 - c2) (b' — c' + ft' - s'h')

+ a (h2 + k2-f d2 — 2ftc)(b' - s'c ') + a ' (h'2 + k'2 + d '2 — 2ft'c') (b - sc)

- f a W + a '2ft'2s + aa ' (b — c + — s (b' — c' + h' - s'ft'} D4

+ {(d2 - c2) (b' - s'c ') + (d '2 — c'2) (b - sc) + a ( + F + d 2 - 2ftc) s'

+ a ' (ft'2 + k'2 + d '2 — 2h'c') s + a (b - c + h- (b' - s'c')+ a ' (b' - c' + - (b - sc)} D3

+ {(d2 — c2) s' + (d'2 — c'2) s + (b — sc) (b' — s'c ') + a (b — c ++ a ' (b' — c' + h' — s'h') s}D2

(36)

S tab ility o f R unn ing 597

+ {(b — sc) s ' -f- (b' —- s'c') s} I) + ss'.



598 F . W . C a r te r .

The quantities Y and G are each involved to the first degree only. The terms involving Y only, omitting the factor Y, are as follows :—

(kaa'2A'2 (A2 + k2) + k 'a 'a2A2 (h'2 + k'2)} D6

+ {k [a'2A'2d2 + aa' (h2 + k2) {h'2 + k!2 + d '2 - 2A'c')]+ k' [a2A2d'2 + aa' (h'2 + k'2)(h2 + k2 + d2 - 2Ac)]} D5

+ {k [a (h2 + k2)(d'2 - c'2) + a 'd2 ( h'2 + + d'2 - 2

+ aa' (A2 + k2)(b' - c' + h' - sT ) + a (b - c)]

+ k' [a' (h'2 + k'2)(d2 — c2) + ad '2 (A2 + + d2 — 2

+ aa' (A'2 + k'2)(b - c + h -s h) + a (b' - c')]} D4

+ {kd2 (d'2 — c'2) + k 'd '2 (d2 — c2)

+ k [a(A2+A2)(b '- s 'c ') + a 'd 2(b '-c '+ A '-s 'A ') +a'(7i'2+A'2+ d '2-2A 'c ')(b-c)

+ k ' [a' (h'2+ k'2) (b—sc) + a d '2 (b—c + A -s +a(A2+A2+ d 2-2Ac) (b '-c')]}D :

+ {k [(d'2 — c'2) (b - c) + d2 (b' — s'c')] + k' [(d2 — c2) (b' — c') + d '2 (b - sc)]

+ k [a (A2 + &2) s' + a' (b — c) (b' — c' -(- A' — s'A')]

+ k' [a' (A'2 + A'2) s + a (b' — c') (b — c + A — sA)]} D2

+ {k [d2s' -f- (b — c) (b' — s'c')] -f k' [d'2s -f- (b' — c') (b — sc)]} I)

+ k (b — c) s' + k' (b' — c') s. (3

The terms involving G only, omitting the factor G, are as follows :—

(ka'A'2 + k'aA2) aa' D(i

+ (k [aa' (A'2 + A'2 + d '2 - 2A'c') + a '2A'2]+ k' [aa' (A2 + A2 + d2 — 2Ac) + a2A2]} D5

+ (k [a (d'2 - c'2) + a' (A'2 + A'2 + d '2 - 2A'c') + aa' ( b '- c '+ A '- s'A')]+ k'[a'(d2 — c2) + a (A2 + A2 + d2— 2Ac) + aa'(b — c + A — sA)]} D4

+ {k[d'2 - c'2 + a' (b' _ c' + A' - s'A') + a (b' — s'c')]-f k' [d2 — c2 + a (b — c + A — sA) + a' (b — sc)]} D3

+ {k [b' — s'c ' + as'] + k' [b — sc + a's]} D2 + (ks' + k's} D. (38)




The terms involving YGr, omitting this factor, are :—{(ka' + k'a) (kaT 2 + k'a/c2) + kk' aa' (h - A')2} D4

+ {ka' [k (A'2 + k'2 + d'2 - 2 h'c') + k' + + d2 - 2A'c)] + k'a [k' (h2 + k2 + d2 - 2Ac) + k (h2 + + d'2 - 2Ac')]} D3

+ {(k + k') (kd'2 + k'd2) — (kc' + k'c)2+ ka' [k (b' — c' + h' - sT ) + k' (b — c + - sh')]-h k'a [k' (b - c + h -s h) + k (b' — c' + - s'A)]} D2

+ {(k + k') (kb' + k'b) — (ks' + k's) (kc' + k'c)} I)+ (k + k') (ks' + k's). (39)

The expressions are given in full, but in most problems simplification is possible depending on the nature of the case. Usually all wheels of a group are alike, so that b — sc = 0 for the group. Terms involving a', which is proportional to the mass of the auxiliary truck, are often small, though not always negligible. The terms of the highest degree seem usually to have little effect on stability. It, however, adds little labour to form the characteristic equation of the locomotive for any particular case, completely, and with all accuracy that the data permits.

We consider first a locomotive of very usual type, having an auxiliary bogie, arranged as shown diagrammatically in fig. 5. The pivotal connection, P, is at the centre of the bogie, in the longitudinal view, and is biassed towards the centre of the transverse view by a force proportional to the displacement; no aligning couple is imposed on the bogie. The characteristic equation for the locomotive is therefore formed from expressions 36 and 37 only. In the example discussed, for which dimensions and other data are given in fig. 5,

2m ----

— l m -22m

4000 kg8000 kg8000 kg8000 kg

Fig. 5.—Bogie Locomotive (4-6-0 or 0-6-4).

the mass of the main frame and parts associated with it, is taken as 60,000 kg., and that of the auxiliary bogie as 4.000 kg., the wheel pressures being as



600 F . W . C a r te r .

indicated in the figure. Following are derived and assumed data on which the characteristic equation has been based.

/ = 2-263 X 10° f = 6-788 X I06 k = 1-473 X 10~7f = 1-131 X 106 f ' = 2-263 X 106 k' = 3k

c = -5 -2 s = 0-0375

b — c = 5-005 d2 = 30-83

d2 — c2 = 3-789

a — 4-5 X H r 4Y2

c' = 0 s' = 0-075

b' — c' = 0

d '2 = 1-5625 d '2 — c'2 = 1-5625

a ' = 0-2 a

h = — 4-16 k' = 0

k2 = 12 = 0 -6h2 + k2 + d2 - 2kc = 16-871 + k’2 + d '2 - =2-162

b — c k — s h — 1-001 b' — c' -f- — s — 0.

The characteristic equation for forward motion, with bogie ahead, is :—

0-288 a4D8 + 5-595a3D7 + [26-135a2 + (0-0240 -f 5-023kY) a3] D6

+ [28-00a + (0-433 + 75 • 74kY) a2] D5

+ [5-921 + (1-564 + 139-6kY) a + (0-901 + 0 • 4806kY) a2] D4 + [65 • 93kY + (1-2815 + 6-856kY) a] D3 + [0-3428 + 7 • 820kY + (0-0751 + 2-211kY)a] D2 + 2 • 488kYD + 0-002812 + 0-3754kY = 0. (40)

At low speed (a = 0) this reduces to5-921D4 + 65 • 93kYD3 + (0-3428 + 7-820kY) D2 + 2-488kYD

+ 0 • 002812 + 0 • 3754k Y = 0. (41)The real and imaginary portions of the roots of the latter equation have been plotted in fig. 6 in order to indicate the degree of initial stability possessed by the arrangement. When Y = 0 the two truck-elements are independent and the roots are given by equation 28. We are thus able to associate one pair of curves (£>q) with the main frame, and the other pair (£V) bogie. The periodic distances for the two elements approach equality for a time, but afterwards diverge, the motion associated with the main frame ultimately becoming aperiodic. For the first portion of the curves, and until kY =0-0211—a fig me which corresponds with a centring force (2Y)




of 2-86 metric tons per cm. displacement—the motion is stable, but beyond this limit it is unstable and the instability is in the motion associated with the bogie. Fig. 7 shows the limiting speed of stable running, deduced from the

002

0 7 2 3 4-

Centering Force.(Tons per c m ctisp/acementj

Fig. 6.—4-6-0 Locomotive. Roots of Characteristic Equation. Zero Speed.

50

4-0

^ 20

X ’ ° ^ 0(

i \! \

/ \ v/ \0 1 2 3

Centering Force (Tons per cm cZ/spiocementJ

F ig. 7.—4-6-0 Locomotive. Speed- limit of Stable Running.

characteristic equation 40. We infer that, beyond the limits indicated in the figures, the bogie tends to lash the rails ; but, being comparatively light, and connected elastically with the main mass of the locomotive, the impacts are unlikely to be a source of danger at ordinary speeds. We reserve further comment until we have considered motion in the reverse direction.

The characteristic equation for motion in the reverse direction, with bogie in the rear, can be written down immediately from equation 40, the quantities b, b', c, c', h, h',being reversed in sign. I t is as follows :—

0 • 288a4D8 + 5-595a3D7 + [26-135a2 + (-0-0240 + 5-023kY) a3]D 6 + [28-00a + ( - 0-433 + 75-74kY) a2] D5 + [5-921 + (— 1-564 + 139-6kY) a + (0-901 - 0-4801cY)a2] D4 + [65-93kY + (1-2815 — 6-856kY)a] D3 + [0-3428 - 7• 820kY + ( - 0-0751 + 2-211kY)a]D2 + 2 • 488kYD + 0-002812 — 0-3754kY = 0. (42)

The region of stable motion now lies between kY = 0 and kY = 0-0075, at which value the independent term in equation 42 passes through zero. The roots of the characteristic equation for zero speed are shown in fig. 8. The motion associated with the bogie preserves its periodic character throughout,



602 F. W . C a r te r .

without any great change in period ; that associated with the main truck increases in period, and becomes aperiodic at about kY = 0 • 0066, where the damping curve splits into two branches, one of which rises rapidly and becomes positive at about kY — 0-0075. The curve of limiting speed for stable running is shown in fig. 9 for this case. We infer that beyond the limit indicated, the

O 1 2 4Centenng force

f Tons per c rnFig. 8.—0-6-4 Locomotive. Roots of Characteristic Equation. Zero Speed.

70

60

r °k 40T 30

I ”g; /oO

/_j////0 7 2 3

Centering foorce (7ons per c/77 d/sp/acernent)

Fig. 9.—0-6-4 Locomotive. Speed- limit of Stable Running.

motion becomes unstable, not by increase in amplitude of a periodic factoi but by a buckling of the wheel-base tending to cause derailment at a forewheel ; and, moreover, that the impacts of the flanges on the rail, when the locomotive is running at speed, are backed by the mass of the main frame and its appurtenances, and are accordingly liable to constitute a source of dangei.

Perhaps the chief conclusion to be drawn from the foregoing discussion is that, for the type of locomotive in question, the centring force for small displacements of the bogie should be moderate in magnitude, and no tendency to lock at the central position should be permitted. This is particularly necessary when the locomotive is intended to run at speed with bogie in the rear. It may be contended that the friction between the parts would be effective in opposing deviation of the trucks ; and in so far as the question is that of running on a perfectly smooth track the contention may be justified. But where a local peculiarity in the track forces the trucks into an unusual configuration, the friction, by opposing their return, may constitute an element of danger. I t is probable that a number of derailments of locomotives having rear bogies which have occurred, and for which no adequate explanation has been vouchsafed, are to be attributed to unstably running locomotives, having




a natural tendency to derail, meeting with some concatenation of unfavourable track conditions.

We next consider locomotives in which the pivot is fixed in both main and auxiliary frames. Chief of these is the locomotive having, in addition to its main axles, a pony axle of the Bissel type ; but more complex machines having the same characteristic are extant, such, for instance, as certain electric locomotives used on the Cascade Division of the Chicago Milwaukee and St. Paul Railroad, in which a six-wheel truck is so pivoted to the main frame. Since the pivot is fixed in both frames, ¥ = 00, and the characteristic equation is formed from expressions 37 and 39 alone.

We take as an example the locomotive shown diagrammatically in fig. 10,

which resembles that of fig. 3, except in that the axle bearing the smaller wheels is carried in an auxiliary frame which is pivoted at P to the main frame. The mass of the auxiliary frame is small and is here neglected, so that a' = 0. The constants involved when small wheels are ahead are as follows :—

-------175 m ------- -----------1 7 5 m -------— 1m — — l m —

8000 l<g 8000 kg 8 0 0 0 kg 6000 kg

F ig. 10.—Bissel Locomotive (2-6-0 or 0-6-2).

/ = 1-96 X 106

/ ' = 1*24 X 10°

f = 5-88 X 106 k — 1-7 X HT7

k' = 8-07 X 1CT7

c = - 2 * 7 5 c' = 1

s = 0-05 s' = 0-0938

b — c = 2-162 b' — c' = -0 -9 0 6

d2 = 10-167d2 — c2 = 2-604

d'2 = 1-5625d'2 — c'2 = 0-5625

a = 5-2 X 10-4 V2

h = — 2

a' = 0

k2 = 4

h? + P + d2 — 2Ac = 7 • 167 m + k2 + d'2 — 2hc' — 13-56.



604 F . W . C a r te r .

The characteristic equation of the locomotive is therefore50• 44a2D5 + [98-0a + ( - 29-25 + 153-lkG) a2] D4

+ [42-56 + ( - 43-42 + 383-7kG) a] D3 + [-16-55 + 239-8kG + ( - 3-937 + 5-355kG) a] D2

+ [2 • 251 + 1 • 325kG] D + 0 • 0509 + 3 • 235kG = 0. (43)For slow motion, the limiting condition of stability requires k G > 0-096,

or a m i n i m u m aligning couple (2G) of about 11 -3 metric tons per cm. movement at a radius of 1 metre. This is impracticably large as a proportional couple throughout the whole range of movement necessary. Fig. 11 shows the roots

A//gn/np Coop/e(Tons per c rr?at 7m TTact/c/s)


of equation 43 when a = 0. I t is not now possible to associate the curves with particular trucks, since the co-ordinates are no longer independent. The curves indicate a simple motion of deviation, always stable, and periodic motion of varying frequency, initially unstable, but becoming and remaining stable ; its ultimate value, when G = oo, corresponds with the case of fig. 3. Fig. 12 shows the limiting speed for stable running as a function of the aligning couple.

For motion in the reverse direction, i.e., with trailing axle, the characteristic equation becomes—

50• 44a2!)5 + [98-0a + (29-25 + 153-lkG) a2] D4

+ [42-56 + (43-42 + 383-7kG) a] I)3 + [16-55 + 239-8kG — (3-937 -j- 5-355kG) a] D2 + [2-251 — l-325kG]D — 0-0509 + 3-235kG = 0. (44)




£ Jo

o /O 20 30 40 so, A/ign/ng Coup/e

( Tons per cm a t

F ig. 12.—2-G-O Locomotive. Speed-limit of Stable Running.

Fig. 13 shows the roots of this equation when a = 0. The fact that stability- does not commence until kG = 0*0509/3-235, is due to the negative sign of the

A/ZgnZng Coop/e( Tons per c m ai / m /?ac//us)


figure 0*0509 in the equation; this figure, however, arises as the difference of two nearly equal quantities and its sign appears to be accidental rather than characteristic of the type. The upper limit of stability is at kG = 1*29, or 2G = 152 tons per cm. at 1 metre radius. At G = 0 the motion has a periodic component, which quickly becomes aperiodic and remains at this until a little after stability has been established, when it again becomes periodic and remains so until G = co. Fig. 14 gives the limiting speed for stable running for this case.

Locomotives of the type now under notice, when run with Bissel ahead, require so strong an aligning couple that it is not unreasonable practice which locks the truck somewhat strongly at its central position. With too small a couple, nosing is likely to become manifest when running at speed. The

2 sVOL. CXXI.— A.



606 F . W . C a r te r .

couple which yields the greatest stability is seen from fig. 12 to be a little greater than corresponds with the lower limit of stability, and this might

. Af/gn/ng Coc/p/e( 7or?s p er Cm

Fig. 14.—0-6-2 Locomotive. Speed-limit of Stable Running.

be obtained by the use of suitable compound springs or equivalent devices. The permissible range of this couple is, however, on account of its high value, so small, that it must usually be surpassed when running on curves and even on straight track the possible deviation of the truck may exceed the permissible range, and become manifest in unstable running. Locomotives of the type are, therefore, rightly considered unsuitable for high speed working. Running the locomotive in the reverse direction, however, the trailing Bissel has a stabilizing effect for a large and useful range of values of aligning couple.

The provision of an aligning couple on an auxiliary truck, in addition to a centring force on its pivot, is not common. I t was proposed by the author for the rear bogies of high speed locomotives,* under the inspiration of some dim and partial apprehension of the performance. I t is employed on the bogies of certain Brown-Boveri-Winterthur electric locomotives recently put into service on the Paris-Orleans Railway, f The author also proposed^ changing the pivotal centres of the auxiliary trucks in symmetrical locomotives of 4-6-4, 4-6-6-4, and similar types, so that the front truck is used as a simple bogie, having such characteristics as shown in figs. 6 and 7, whilst the rear truck trails, and has general characteristics such as shown in figs. 13 and 14. I t was the intention that the locomotive should run equally in either direction,

* See British Patents 128106, 155038.t Parodi, “ Electrification partielle du reseau de la Compagnie d’Orleans, ’ ‘ Revue

Generale des Chemins de Fer,’ 1927, 46e annee, 2me semestre, p. 23, fig. 2.I See British Patent 163185.



Stability o f Running o f Locomotives. G07

the change of pivots being brought about by the reversing gear as part of the control system. Very stable running would be expected of such a locomotive. Arrangements for changing the pivotal centres were provided, as an experimental measure, on one of the Brown-Boveri-Winterthur locomotives referred to above, but the results of tests have not yet been disclosed.

Fig. 15.—Double Bogie Locomotive (4-0-4).

A common type of locomotive of two axle groups is the double bogie locomotive shown diagrammatic ally in fig. 15 ; and the bogie coach, whether motor or trailer, may be included in the same category for the present purpose. This, since it involves three parts, the two bogies and the body, leads in general to a characteristic equation of the twelfth degree,* which, however, in the case of a symmetrical locomotive, is readily split into two of the sixth degree, as will be shown. We consider the symmetrical case only.

We take the centre of each bogie as reference point for its own motion, distinguishing between them by suffixes 1 and 2, and the centre of the body as reference point for its motion, using co-ordinate without distinguishing marks. Let M be the mass of the body and I its moment of inertia about the vertical axis through its centre. Let 2s be the distance between bogie centres. For the bogies, with the usual notation, b = 0, c = 0, 0 ; also

ai = a 2 ( = a)> <*i = <M = d), si — s2(— s), k, = k2(= k ) , k1 = k2(= k).From equations 23 and 24,

(al)2 -f- D) — ik Y /, % (45)(aD2 + D) - <l,2 = |k Y 2', (46)

sy1 -)- (a&2D2 -f- d2D) <jq — 0, (47)sy2 + (a&2D2 -J- d2D) = 0 ; (48)

alsoMV2D2?/ = - (Y/ + Y2'), (49)IV2D2 = —* (Y/ - ; (50)

and we may writeY / = - 2Y (j,, - y - (51)Y2' = - 2Y (y, - y + 4 ) . (52)

* See § 9 below.



♦508 F . W . C a r te r .

Adding equations 45 and 46, 47 and 48, and substituting in the results, and in equation 49. for Y / Y2' gives

(aD2 + D + kY) (yx + y 2) — + <{/2) — 2kY — 0, (53)

s (th + y2) + (aA-2D2 + d2D) (<{q + ^2) = 0, (54)2Y (yr + y2)- (MV2D2 + 4Y) = 0 ; (55)

writing aa = JkMV2 these equations yield

a [a3/c2D° + a2 (Z;2 + d 2) D5 -f ad2D4 + ash)2]

+ kY [(1 + a) a2&2D4 + a (1c2 + d2 + ad2) D3 + d2D2 + s] = 0. (56)

This is a partial characteristic equation of the locomotive, and, since there is no term in D, it represents an unstable motion. In a similar manner, by taking the differences of equations 45 and 46, and of 47 and 48, substituting for Y / and Y2' in these, and in equation 50, the remaining components of the motion >re found. The resulting equation is similar to equation 56, with (3 written in place of a, where (3a = JkIV2/s2, and therefore again represents an unstable motion. The destructive effective of the instability is, however, limited on account of the comparatively small mass of the trucks.

9. Locomotives of more than Two Axle-Groups.The general locomotive-unit may be said to comprise three axle-groups,

a main group, with auxiliary groups fore and aft of it. The terms “ main ’ and “ auxiliary ” must be considered as having reference to the frame through which draught is effected rather than to primary function : for, although in the steam locomotive the main axle-group comprises the driving axles, the auxiliary groups being used for steering and weight-distributing purposes, in the electric locomotive there is no such functional distinction, and auxiliary groups may consist partly or wholly of driving axles ; indeed, in the doublebogie locomotive (4-0-4) the main group is lacking. A complete locomotive may consist of two or more units, but in such locomotives some of the intermediate auxiliary axle-groups are often omitted. Thus the locomotive usually represented by the symbol 4-6-6-4 is a two-unit locomotive lacking in two auxiliary axle-groups, and might with propriety, certainly with increased lucidity, be represented by the symbol 4-6-0 + 0-6-4. When two units are connected by means of a draw-bar, they are, for first order deviations, independent of one another ; but when the connection takes the form of a Mallet hinge the locomotive must, for the present purpose, be considered




as a whole. The hinge coupling is, however, unusual for locomotives intended to run at high speeds.

Each axle group in general adds four to the degree of the characteristic equation, but each fixed pivot or hinge joint reduces the degree again by two. Thus the locomotive 4-6-6-4, if the units are connected by a draw-bar, has a characteristic equation of the sixteenth degree, which, if the units are similar and the action of intermediate side buffers is neglected, can be divided into two equations, each of the eighth degree, related to one another in a manner similar to equations 40 and 42. If, however, the units are connected by a hinge, the characteristic equation of the locomotive is of the fourteenth degree. I t is apparent, therefore, that the characteristic equation increases rapidly in length and complexity as the number of axle-groups is increased. No difficulty arises in forming the equations of deviation for any locomotive, but the deduction therefrom of the characteristic equation involves considerable labour when the number of axle-groups exceeds two.

It is not the author’s purpose, in the present communication, to discuss the calculation of more complex cases further. Sufficient has been said to demonstrate the proposed treatment of the problem of stability, and the methods for deducing the characteristic equation of the locomotive, on which stability depends. The general effect of certain usual wheel arrangements has been discussed with the aid of particular examples ; and from such partial studies the general performance in more complex cases may be inferred. Certain of these cases are perhaps worthy of a more complete treatment, which may be accorded them hereafter as occasion demands.

A p p e n d ix — N o ta tio n .

The following notation is used in the paper—dashes and suffixes being used to distinguish symbols of a similar nature :—

a, radius of wheel, in millimetres in equations 1 and 2, elsewhere inmetres.

b, a half the distance between the treads of the two wheels of a pair—a little more than a half the gauge of track ; taken as 0-75 m. for standard gauge.

x, y, co-ordinates of a selected reference point on the longitudinal axis of symmetry of the locomotive or truck plan, referred to fixed axes ; x is measured in the direction of the track, and from the centra line of the track.

c, distance of an axle ahead of the selected reference point ( ).



610 Stability o f Running .

h,distance of the centre of inertia of a truck ahead of the selected reference point ( xy).

s, distance of a reference point for outside force system ahead of the selected reference point {xy).

f, creepage coefficient, defined and explained in section 3.Je,radius of gyration of a truck about a vertical axis through its centre of inertia.

m, M, masses, generally of truck or other assembly of parts associated in movement. [N.B.—Since forces are herein taken in gravitational units (kg.), inertial masses must be divided by g, see a below].

w, weight borne by a wheel (kg.). mk2, 1, moment of inertia of truck, or other assembly of parts.

X, component force on truck in direction of track, also used for periodicdistance.

Y, component force on truck transverse to track.G, couple acting on truck about a vertical axis.V, velocity of progression of locomotive, metres per sec.

a, [j , ratios, explained in section 8.y, creepage, defined in section 3.X, conicity of wheels, generally 1/20.d», angle between axis of wheel-base of locomotive or truck and line of

track.

In the following, the summation is .taken over the axles of an associated group :—■

f = S ( / ) , k = 1 /S (/) , a ^ m V ^ S (/) , b = S ( / X 6- o ) / s (/),\ d l l

c = S(/c)/S (/), d2 = 6* + S(/c2)/S (/), s = S ( " / A - ) / z ( / ) .\ w /

Here a, b, c and d are linear magnitudes, but s is a number, and f a force. When, as is usually the case in practice, the wheels of an axle-group are of

equal radius and carry equal loads,/is the same for all axles and the expressions given above are simplified. Thus, if n is the number of axles in the group,

f = nf, c = - 2(c), s = X -, and so on. n a

We may note the following expressions of frequent occurrence :—

b - sc = X62 f r f s ( Cr~ C8)ar



Photoelectrons produced by X-Rays. 611

This, a function of the differences of the c’s, is independent of the selected reference point. I t is zero when the wheels of the group are of equal radius, whether or not they are equally loaded.

d2 _ C2 = 62 + 2 {Cf _ Cs)2]/[2 {f)f}

h2 + k2 + d 2 — The = k2 -j- 62 + S [ / (h — c)2]/2 (/),A2 + & + d'2 - 2hc' = B + 62 -f S [ / ' - c')2]/E (/').

These expressions are also independent of the choice of reference point.

The Spacial Distribution o f Photoelectrons produced byX-Rays.

By E. J. W illiams, J. M. N uttall, and H. S. B arlow.

(Communicated by W. L. Bragg, F.R.S.—Received August 7, 1928.)

Introduction.Investigations on the initial directions of emission of photoelectrons produced

by X-rays show that the distribution depends on the direction of propagation of the rays, whilst in the case of a polarised beam it depends also on the direction of the electric vector of the incident radiation. The distribution of the initial directions of the photoelectrons relative to the direction of the incident X-ray beam is usually called the “ longitudinal distribution,” whilst that relative to the plane containing the electric vector and the rays, in the case of a polarised beam, is termed the “ lateral distribution.”

The present paper will deal only with the longitudinal distribution of the photoelectrons. In the first place a survey of the experimental situation will be given, from which it will be seen that the results of previous observers are not in a satisfactory state of agreement. Secondly, it is the purpose of this paper to give an account of the authors’ own experiments on the longitudinal distribution in which it is hoped that some of the possible errors giving rise to the discordant results of previous investigators have been eliminated. The main interest of the present work lies in the results obtained for the asymmetry, which show that the average forward momentum of a photoelectron is nearly



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