On the Geometrical Properties of the "Atriphthaloid"

On the Geometrical Properties of the "Atriphthaloid"Author(s): Richard TownsendSource: Proceedings of the Royal Irish Academy. Science, Vol. 4 (1884 - 1888), pp. 62-70Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20635910 .

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62 Proceedings of the Royal Irish Academy.

VII.?On the Geometrical Properties of the " Atbiphthaloid."

By the Hey, Eichard Townsekb, ILA., F.T.C.D.

[Read, February 27, 1882.]

Dr. Hatjghton has discovered the existence of a large family of curves in the course of his investigation of the form of a frictionless ocean, covering an attracting sphere.

The whole family he proposes to call Atriptotkalassic curves, but one of them is so simple and elegant, that he has named it the

Atripkthaloid, and requested me to undertake its discussion, which I have done in the following Paper:?

The equation of the Atripkthaloid in polar co-ordinates is

r*(A -

2gr) = cz cosec20.

Putting for convenience of discussion, A = 2gk, c1 -

2g$, and 0 = 90? - w, the equation of the curve assumes the form?

r~ (r - h) + ?3 sec2 w = 0, (1)

where h and h are positive constants representing linear magnitudes, which may have any independent values from 0 to go, and which may be regarded as the parameters of the curve.

The equation (1) giving the same values for r when w is changed into -

o), or into ir ? to, we see at once that the curve it represents is

symmetrical with respect to the two rectangular axes o> = 0 and q> = \ it, and has a centre at the origin 0 (see fig.)

The equation (1) heing, for every value of <*>, a cubic in r whose absolute term is positive: hence, for every value of o>, r has one real

negative value, commencing from its minimum absolute value OC when <*> = 0, and increasing continuously to go from o> = 0 to o> = ^7r. Hence (see fig.), the two symmetrical conchoidal-shaped infinite branches, which always meet asymptotically on the axis for which ?) = i tt, and which never disappear for any finite values of h and h however related to each other.

The remaining two roots of the cubic (1) for r in terms of o> being real or imaginary, by the theory of equations, according as sec2 cd is <

4 ?2 4 hz 01 >

27 & 9 &ence> when ?

^ is > 1, the curve (see fig.), in addition

to the two infinite symmetrical conchoidal branches which never dis appear, has two finite symmetrical ovals, lying entirely outside the conchoidal branches, and intersecting the axis of x at two pairs of real



Town send?Geometrical Properties of the Airiphthaloid. 63

points A and B. The two tangents OE and OF to which from the 4 ]p

origin 0 are given in position by the equation sec2 w = - and in

2 magnitude by the equation OE = OF -

o A, and the equal distances ?

OB of their chords of contact EF from the origin by the equation u

h 4 /&3

"When ?

jj =

1, then for the two tangents sec2 o> = 1, and the ovals

consequently contract into points whose equal distances from the ori 2

gin = - h: the pairs of vertices A and B of the two ovals (see fig.) then o

coincide, and the equal distances OC of the two conchoidal vertices from

the origin = -// .

When ^ ^

is < 1, then for the two tangents sec2 w < 1, and there

fore w is imaginary. The two ovals then disappear altogether, and the curve consists entirely of the two conchoidal branches, which never dis

appear. The sum of the three roots ru r2, r^ of equation (1) being indepen

dent of the value of oo, and = h for all directions of r; hence, for the



64 Proceeding a of the Royal Irish Academy.

two tangents, since rx = r2 = - h for the ovals, therefore r3 =

- - A for

the conchoidal branches. Hence (see fig.) the tangential radii OR and OF of the ovals, when real, are bisected at IT and IT by the conchoidal branches at their sides of the asymptotic axis of the curve.

The sum of the three products in pairs r2r:] + r^rY + i\r% of the three

roots rh r2, r3 of equation (1) being also independent of o>, and equal 0

for all directions of r; therefore, for every three radii of the curve

having a common direction, if rY and r2 be those to either oval, and rs that to the infinite branch at the same side of the asymptotic axis, ?*3 + r2)

= riT2. Hence, for every three radii having a common

direction, that to cither infinite branch is in magnitude and direction half the harmonic mean between those to the oval at the same side of

the asymptotic axis of the curve.

Differentiating equation (1) with respect to <*>, we get

*L=-.2&. ? . -1? (2) rdbj

~ * ' cos'to

' 3r - 2h

' r~*

dr which shows that is = 0 only when sin w = 0 : that is, only for the

three apsidal points A, R, C of the curve. Hence the radius r of the curve has its maxima and minima values

only at the three apsidal points on the axis for which w = 0, and the conchoidal branches have in consequence no dumb-bell depressions throughout their whole lengths.

Since, from the same equation (2), ~ = oo when w = i tt and when

3r - 2h = 0 ; therefore, as already observed, the axis for which w = i ir

is an asymptotic tangent to the infinite branches, and the two radii OF and OF for which 3r = 2A are the two ordinary tangents to the ovals from the origin.

Putting, in equation (1), x = r cos oj and y = r sin o, that is trans

forming into rectangular co-ordinates, and solving for y, we get

f = h? - x2 - 2MV2 + k?x\ (3)

which shows that, for every finite value of x2, y~ has but a single finite value, positive or negative; that, from x - oo to x =

OA, y1 is nega

tive, and therefore y imaginary; that, from x = OA to x - OB, y2 is positive and finite, and therefore y real and finite; that, from x - OB to x = OC, y1 is negative, and therefore y imaginary; and that, from x = OC to x = 0 y2 is positive and increasing from 0 to oo, and there fore y real and increasing from 0 to od. These results obviously verify the form of the curve as obtained above from the equation for r (1).

Putting ;/ - 0 in equation (3), we obtain for the squares of the three



Town send? Geometrical Properties of the Atriphthaloid. 66

semi-axes a, b, c of tlie curve, that is of the three distances OA, Oil, OC of the three vertices A, B, from the origin 0, the cubic equation

x6 - + 2hkV - k* = 0, (4)

which has always one real positive root, and whose three roots when real are all positive.

N.B.?The equation (4) is, by the theory of equations, that whose roots are the squares of those of equation (1) for the case when o> = 0.

Since from equation (1) when a> = 0, by the theory of equations,

a -f b + c - h, be + ca + ab = 0, abc = - ?:{

therefore

ab ?l i ab vir ?l b2 c1 =-, h =-?, = ?

(o; a v o a + b a + b

and if d denote the distance OB (see fig.), since d2*= 3 ?, therefore also

d2 = 3 ~??f~?? relations which give the values of the four quanti az h ab f bl

1

tics c and d, h and k, in terms of a and b, and show consequently that the two latter quantities, which when real may have any independent values from 0 to oo, determine completely all the particulars of the curve in every case.

In the particular case when 4 A3 = 27k3, that is when the two ovals contract into points, and when consequently a

= b, equation (4) has 4 1

two of its roots each = - /r, and its third root = - h2.

In the particular case when li = 2h, in which as 4A3 > 27X*3 the two ovals are real, the three roots of equation (4) are respectively xl =

k'1,

and (3 ? V5) k\ In the same case, those of equation (1), when w -

0, are respec

tively r = k, and r - \ (1 ? Y5)?; as they ought, the roots of equa

tion (4) being in every case the squares of those of equation (1), when in the latter oj = 0.

Putting in for h and k in the function (4A3- 27P) ?, on whose sign as positive or negative it depends whether the ovals are real or

imaginary, their values in terms of a and b as given by equation (5), we find readily that

(4?3- 27#)?i = '(2(?3 + ab + L*)+ 3ab)ab

(a -

bf, (6)

and that its sign is consequently, as it ought to be, positive or nega tive according as a and b are real or imaginary.



Proceeding* of the Royal Irish Academy.

N.B.?The function in question being, by the theory of equations, the product of the squares of the differences of the roots of equation (1) for the case when o> - 0, the same value for it in terms of a and b would

be obtained even more readily than above by the substitution for c in that product of its value in terms of a and b. See equations (1) and (5).

Multiplying both sides of equation (3) by irdx, integrating between the limits a and b, and substituting in the result for h and h their values just given in terms of a and 1; we get, for the volume Fof the solid generated by the revolution of either oval round its axis of figure

AB, the value in terms of a and b, viz.?

which, compared with that of the volume 8 of the sphere on AB as diameter, viz., I tt (a

- bj\ gives for the ratio of the two volumes in

terms of a and b the value

?=4^-;*, (8)

a value which, lying always between the extreme limits 3 and 4 cor

responding respectively to the extreme values 1 and a> of the unre

stricted ratio of a to b, shows, consequently, that the extreme depth MN is always greater than the extreme breadth AB of the ovals (see fig.)

That the chords of contact EF of the tangents to the ovals from the centre 0 of the curve (see fig.), which when the ovals are finite are of course always less than their extreme depths MN, are also in all cases greater than their extreme breadths, AB} may be readily shown from equation (6) as follows. Since

/ 4 P\ 4 /4?3 _ 9 7/-3

9 hj <H //

therefore from equation (6), as (a-b) = AB, and as hi? = a~b~(az -vab + b~) ~ (a + b)~ by equations (5),

EF f 4 [2 (tf- + ab + V) + SabJ AJjj

~ 9 ^r^abT?)(a + b)r' ̂ J

a ratio always exceeding unity, und having the extreme values 27 -f- 9 and 16^9 for the extreme values 1 and oo of the unrestricted ratio of a to b.

The value of MNm terms of a and b not being in general deter minate like that of EF in finite terms, the exact value of the ratio of



Townsend?Geometrical Proper ties of the Atripht haloid. 67

MN to AB cannot consequently be given in general in such terms.

As PF h never greater than J/^when both are real, and only equal to it when both are evanescent, the least possible value of the ratio of

PFto AB given by equation (9), viz., 4 -f 3, is consequently an in ferior though by no means a close limit to the possible ratio of

MN: AB. But a superior as well as a closer inferior limit, both

however much outside the extreme limits of the actual ratio, may be

found for it from equation (7) as follows. As the volume V o>i the solid generated by the revolution of either

oval round its axis of figure AB must necessarily be less than that of

the circumsc ribed cylinder, and greater than that of the inscribed double cone having AB for axis and the circle described by MNior base, we

must therefore have, for all values of a and b,

h^\a* -p)<i7T. MN2. AB > -A- TT. MN2. AB, \a 4- b

and therefore, for all values of a and b, as (a -b) = AB,

MAT\2 na2+ ab + ? a a2 + ab + b2 . x

ab)k8 T?W > 1

-j^TTf-' '10}

from which it follows, consequently, that the square of the ratio in question must always lie between the extreme limits 2 and 8, its least and greatest values corresponding respectively to the least and greatest values 1 and go of the unrestricted ratio of a to b. Prom the manner in which they have been obtained, however, these limits are obviously

much outside of those of the actual ratio.

Differentiating equation (3) with respect to x, we get

u%=-x\ 2M<V3 - 2k6x-5, (11) dx

v ;

from which we get, for the values of x~ for which == 0 and for which

consequently y2 is a maximum or a minimum, the cubic equation

xC) ? 2hhzx2 + 2kQ = 0, (12)

which has always one real negative root for which x is consequently imaginary, and whose remaining roots will, by the theory of equa tions, be real or imaginary according as 8 A3- 27 A*3 is > or < 0, and of the same sign when real. Hence the curve has never more than two values of x2 for which yl is a maximum or a minimum; and as those values of x2 lie necessarily, one between the limits a2 and b2 for which y2 is positive, and the other between the limits b2 and c2 for which y2 is negative, there is therefore never more than one pair of maxima

11. I. A. PltOC., SEIL Tl., VOL. IV.?SCIENCE. i,




double ordinates to tbe curve which are real as regards both position and

magnitude, viz., those MN of the ovals when real (see fig., page 63). In the particular case when 4A3 = 27#*, that is when the two ovals

contract into points, the three roots of equation (6) are, respectively,

*2 = ^2,

^ = ?(Y3-1)#, and = - ~(V3 + \)h\

the first only of which gives a pair of chords real as regards both posi tion and magnitude of the curve.

Eliminating x2 between equations (3) and (12), we get, after a few

ordinary reductions, for the actual maxima and minima values of y21 the cubic equation

4y* -

Shy + 4? (h3 - 9#) y* + & (4h3 -

2W) = 0, (13)

which has always one real positive root corresponding to the real negative root of equation (12), and belonging consequently to no chords real even as regards position of the curve; and whose remain

ing roots will be real and have opposite signs when 4A3 - 27?3 is posi tive, that is, when the two ovals are real and finite.

In the particular case when 4A3 = 27A3, that is, when the ovals contract into points, one root of equation (13) is evanescent, and cor

responds to the evanescent ovals; and the remaining two are respec 2 -

tively = W (1 ? -

V3); which correspond respectively, the former to the o

real negative root of equation (12) and therefore to no real chords even as regards position of the curve, and the latter to the pair of chords between the evanescent ovals and the infinite branches, which, though real as regards their positions, have no real intersections with the curve.

Differentiating with respect to x the value of ~ given by equa CLX

tion (11), having first substituted in it for y its value in x given by equation (3), and equating the result to 0, we get, after a few ordi

nary reductions, for the values of x% for which ~ 0, that is, for the dx

several points of inflexion, real or imaginary, of the curve, the equa tion of the sixth degree,

x2 [h2x^ - Wh&x8 + 3P (2h* + 5P) x& - 18**?*

+ lShk9x2- 6?12]=0, (14)

of whose roots one is obviously evanescent, one essentially real and

positive, and the remaining four when real essentially positive also. The evanescent root corresponds of course to the axis of y} which

is consequently a doubly inflexional as well as asymptotically tangen



Townsend?Geometrical Properties of the AtriphthaloicL 69

tial chord of the curve, and the essentially real and positive root to the pair of inflexional chords PQ (see fig.) of the conchoidal branches; which, equidistant in opposite directions from the axis of y> lie neces

sarily somewhere between it and the parallel tangents at the vertices C of the branches.

The four remaining roots, with the four pairs of inflexional chords reflexions of each other with respect to the axis of y to which they cor

respond in the curve, may be all real, or all imaginary, or two real and two imaginary, according to the particular value of the parametric ratio h ~ k on which alone they depend. When the two ovals are real and

finite, as they are for all values of the ratio for which 4hz > 27F, the entire number of pairs of inflexional chords intersecting them at pairs of real points is necessarily even; but none of the four pairs, even

when themselves real as regards their positions, need intersect them

necessarily at real points at all. So that the ovals may be, and in fact often if not always are, as represented in the figure, concave

to their interiors throughout the entire circuits of their perimeters. The application of Sturm's theorem to equation (14) gives us, for

all numerical values of the parametric ratio, the exact numbers of

corresponding pairs of inflexional chords which occupy real positions within any two assigned limits of distance from the centre 0 of the curve; and, as the corresponding values of OA and OB can also be determined for all such values to any degree of approximation, from equations (1) or (4), by Homer's and other methods of numerical solu tion, the exact numbers of real pairs of inflexional chords lying within the intervals AB, and therefore intersecting the ovals at real points, can consequently be determined by its aid for all numerical values of the ratio. Its applications, however, are in general laborious, and in the present instance uninstructive except for such values of the ratio.

By its application to the equation for the two particular cases when h -T k = 1 and 2 respectively, which correspond, the former to an imaginary and the latter to a real pair of ovals; we find, with comparatively little trouble in either case, arising from the circum stance of the quadratic functions having imaginary roots in both, and therefore dispensing with the necessity of proceeding any further with the process in either, that the equation, in addition to its evanescent, has for each of them but a single real root, that, viz., corresponding to the pair of inflexional chords of the infinite branches which are always real. And, by a similar application, attended with a little more trouble arising from the reality of the roots of the quadratic functions in each case, we arrive at the same result for the two cases when 4h3 = 26k3 and 28^ respectively, which correspond again, the former to an imagi nary and the latter to a real pair of ovals; the intermediate case for which 4A3 = 27#* being that for which they pass through evanescence from their real to their imaginary state, and conversely.

In the particular case when 4hz ~ 27/^, for which the ovals contract into points, the five finite roots of equation (14) are on the contrary




all real, and correspond in consequence to five pairs of chords all real as regards their positions; one of which intersects, as in all cases, the

conchoidal branches, and, of the remaining four, three coincide at the

evanescent ovals, and the fourth intersects the curve at imaginary

points. For, equation (14) for xz is easily seen to be equivalent in that

case to

which, besides its single root = 0, has evidently three roots each - 7r,

and two others equal respectively to

which correspond respectively, the first three to the evanescent ovals, the latter with the lower sign to the conchoidal branches, and the latter with the upper sign to no real points at all on the curve.

Of the three pairs of chords coinciding at the evanescent ovals in this case, two however correspond to the evanescent radii of curvature

at their vertices; the function - yz ̂ ^, which multiplied by xl? is mani

festly equivalent to the quantity within the brackets at the left side of equation (14), representing in all cases, as is well known, at the several vertices of any curve symmetrical with respect to the axis of x, the

squares of the corresponding radii of curvature at the vertices, and

being consequently evanescent at the two coincident vertices of every acnodal double point on the axis of the curve.

As regards the third pair in the same case. Taking it in con

nexion with the pair intersecting the curve at imaginary points, and

conceiving both pairs to change position together with the gradual and continued increase of the parametric ratio from its critical to every higher value, and the consequent accompanying dilatation of the ovals from their evanescent to every greater magnitude; they are to be re

garded, while real, as two variable pairs of inflexional chords inter

secting the expanding ovals at pairs of imaginary points, and after

coming together in the course of their variation, as the above particu lar cases show they do very rapidly with the increase of the ratio, then passing through coincidence from their real to their imaginary state,

beyond which the particulars above stated supply no clue to follow them.

N.B.?The questions, as to whether for any values of the parame tric ratio the ovals have ever real points of inflexion, and as to the critical values (if any) of the ratio for which they cease (if ever) to be, as represented in the figure, concave to their interiors throughout the entire circuits of their perimeters, have, it will he observed, been left undecided in the above investigation.

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On the Geometrical Properties of the "Atriphthaloid"