Seminvariants of a General System of Linear Homogeneous Differential EquationsAuthor(s): E. B. StoufferSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 6, No. 11 (Nov. 15, 1920), pp. 645-648Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/84227 .

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VOL. 6, I920 MATHEMATICS: E. B. STOUFFER 645

SEMINVARIANTS OF A GENERAL SYSTEM OF LINEAR HOMO- GENEO US DIFFERENTIAL EQUATIONS

BY E. B. STOUFTER

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KANSAS

Communicated by E. J. Wilczynski, September 27, 1920

The system of linear homogeneous differential equations m- 1 n

yim + i ()P ?klYk() =

O, (i =

1,2, . ,n), (A) 1=0 k=l

where

y(l) = dyk Yk dx

and where Piki are functions of the independent variable x, is evidently converted into another system of the same form by the transformation

n

Yk = akyx (k = 1,2, ... ,n), (1) X=i

where ankx are arbitrary functions of x, and where the determinant 11 ak= A does not vanish identically. Furthermore, it is known' that (1) is the most general transformation of the dependent variables which leaves (A) unchanged in form.

A function of the coefficients of (A) and their derivatives which has the same value for (A) as for every system derived from (A) by a trans- formation of the form (1) is called a seminvariant. The seminvariants of (A) have been calculated for the case2 n = 1, m = any positive integer, and for the case3 m = 2, n = any positive integer. It is the purpose of the present paper to obtain them for the general case of (A).

The calculations are considerably simplified by first obtaining the seminvariants in a so-called semi-canonical form and then expressing them in terms of the coefficients of (A). The possibility of a simplification of this general nature was first suggested by Green.4

The transformation (1) converts (A) into a new system (B) in which the coefficients of the derivatives of order m - 1 of the dependent variables are zero provided that aij is so selected that

n n

aij = -- Pi,k,m-l1ikj = E qiklakj, (i,j = 12, ....,n). (2)

k=l k=l

Such a selection is always possible. The system (B) is the semicanonical form of (A).

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646 MA THEMATICS: E. B. STOUFFER PROC. N. A. S.

When the coefficients of (1) are subject to the conditions (2) it is easily proved by induction that

n

= q C qikTakj, (3)

k=l

where n

qikt -

qi,k ,r-- + - qi,X,-- lqXkl.

X=I

If rikl denotes the coefficient of (B) corresponding to Piki of (A), we find

by straight substitution n

&kZ = E ^-Ax 0i ak-I) +

X=l n m--X--

2 E ( m l ) Px i+ -a)] (1 = 0,1,.. .,m - 2), (4)

p=1 "=0

where Axi is the algebraic minor of axi in A. By the use of (3) the expres- sion (4) for rikl may be put into the form

n n

'7ikl = E ^Axia, kUX, (5)

X=l A=l

where n m-X-I

ui =l - qx, A, m-I j - qX, v, m-l-TrqpvTr.

Y=1 7=0

Again, we find by differentiation n

Arikl = A xi AakvXl., (6) X=in ^11

where

Vp.l = uX,1 + j (quluXlp q-hlvyl)-

v=l

Similarly n n

Akr ikl = A ANxiakWXz. (7) X=l M=l

where

Wxty =: vX1I + - (q,,g^lvh - qx,,lVi)- V=1

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VoL. 6, 1920 MATHEMATICS: E. B. STOUFFER 647

From (2) it follows that the most general transformation of form (1) which leaves (B) in the semi-canonical form is given by

n

Yk = akya , D = lakxl 0, (k = 1,2, ....,) (8) X=I

where akx are arbitrary constants. Equation (4) shows that such a transformation converts (B) into a new system whose coefficients 1ikl and their derivatives 7rk) are given by the equations

n n

D(k = A x7r(T a,k, (I = o0, ,m - 2), (9) X=1 A=1

where Axi is the algebraic minor of axi in D. If the transformation (8) is made infinitesimal, it is found that all

seminvariants in their semi-canonical form must satisfy the system of partial differential equations

oo n

2(7rkrl (r) - 7s ) O7rksl

r=o k=l

(r,s= ,2 ...n;l=0,1,2. . .,m -2). (10)

For r = 0, I = m - 2, there are n solutions given by3

n a7r1, 1, m-2 7rl, 2, m--2 * .*. 71, n, m- 21

r(roo ) ( 1 2, i, m 2 2,2, m-2 ....* 2, , m-2

r! \ 7ri, i,m - 2/ .............................. ti=l

7rn, 1, m -2 7rn, 2, m--2 - 7rn, n, m - 2

(r = 0,1,...,n - 1) For r = 1, 2, 1 = - 2, solutions are given by5

n n

(r so) - I I(o) - 1 (7ri,j, m-- 2 j, i , (r + s < n),

1/ 7r,-

(r) ri_ m 2 (r s o)

tI , j (6-- x 2 )ri,ij, m-2 i=1 j=i1 (r + s < n; t = 1,2,3; t < s).

For = 0, I = 0, 1, .... ,m - 3, n2 solutions for each value of I are given by

n n

t! = r ( b Y-Irso) , r 0(+ s < n t=1,2; .t <;.

i-= j=1

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648 PHYSICS: A. G. WEiBSTER PROC. N. A. S.

The above seminvariants may be converted into seminvariants of (A) by means of equations (5), (6), (7). A comparison of these equations with (9) shows that the desired seminvariants of (A) are obtained simply by replacing in the above semi-canonical forms rikt by uikl, 7kl by Vikl and 7rikl by wikl.

A comparison of the seminvariants I(r St) of (A) with the corresponding seminvariants5 for the case n = 2 shows the former to be independent. Moreover, the functional determinant of 11tst) with respect to .ikl for ;each value of I = 0, 1, 2. . .,m - 3 shows that Ist) are independent among themselves and of the seminvariants I(rst). Equations (10) show that we have the proper number of solutions for the variables involved and that all other seminvariants of the complete system can be obtained

by the differentiation of I(r s ) and Ir t). We have therefore the following theorem:

All seminvariants of (A) are functions of I(rst) (r = 0,1..,n - 1; r + s <

n; t = 1, 2, 3; t < s), It) (r = O, . .,n - 1; r + s < n; t = 1, 2; t < s; I = 0,1,. .m - 3), and of the derivatives of I(rst) (t = 1,2) and I1(t).

1Wilczynski, E. J., Projective Differential Geometry of Curves and Ruled Surfaces, Teubner, Leipzig, Chap. I.

2 Wilczynski, E. J., Ibid., Chap. II. 3 Stouffer, E. B., London, Proc. Math. Soc. (Ser. 2), 15, 1916 (217-226). 4 Green, G. M., Trans. Amer. Math, Soc., New York, 16, 1915 (1-12). 5 Stouffer, E. B., London, Proc. Math. Soc. (Ser. 2), 17, 1919 (337-352).

SOME NEW METHODS IN INTERIOR BALLISTICS*

BY ARTHUR GORDON WEBSTER

CLARK UNIVERSITY, WORCESTER, MASS.

Read before the Academy, April 26, 1920.

The principal problem of interior ballistics is, given a particular gun, a particular shot, and a particular kind of powder, to find, for a given load, the position and velocity of the shot, the mean pressure (and inci-

dentally temperature) of the gases in the gun, and the fraction of the

powder burned, all as functions of the time or of each other until the exit of the shot from the muzzle of the gun. In particular, we wish to

know the muzzle-velocity of the shot, the maximum pressure to which the gun will be exposed, and the portion of the bore which will be exposed to it. It is then the duty of the mechanical engineer to design a gun to

safely resist the pressure that may be expected. Or it is the inverse

problem of the ballistician, by experiments on the action of the powder in the gun, to find its properties and those of the gun. It is true that the

properties of the powder are more conveniently studied by means of

*Communication from the BAALISTIC INSTITUTE, CLARK UNIVERSITY, No. so.

648 PHYSICS: A. G. WEiBSTER PROC. N. A. S.

The above seminvariants may be converted into seminvariants of (A) by means of equations (5), (6), (7). A comparison of these equations with (9) shows that the desired seminvariants of (A) are obtained simply by replacing in the above semi-canonical forms rikt by uikl, 7kl by Vikl and 7rikl by wikl.

A comparison of the seminvariants I(r St) of (A) with the corresponding seminvariants5 for the case n = 2 shows the former to be independent. Moreover, the functional determinant of 11tst) with respect to .ikl for ;each value of I = 0, 1, 2. . .,m - 3 shows that Ist) are independent among themselves and of the seminvariants I(rst). Equations (10) show that we have the proper number of solutions for the variables involved and that all other seminvariants of the complete system can be obtained

by the differentiation of I(r s ) and Ir t). We have therefore the following theorem:

All seminvariants of (A) are functions of I(rst) (r = 0,1..,n - 1; r + s <

n; t = 1, 2, 3; t < s), It) (r = O, . .,n - 1; r + s < n; t = 1, 2; t < s; I = 0,1,. .m - 3), and of the derivatives of I(rst) (t = 1,2) and I1(t).

1Wilczynski, E. J., Projective Differential Geometry of Curves and Ruled Surfaces, Teubner, Leipzig, Chap. I.

2 Wilczynski, E. J., Ibid., Chap. II. 3 Stouffer, E. B., London, Proc. Math. Soc. (Ser. 2), 15, 1916 (217-226). 4 Green, G. M., Trans. Amer. Math, Soc., New York, 16, 1915 (1-12). 5 Stouffer, E. B., London, Proc. Math. Soc. (Ser. 2), 17, 1919 (337-352).

SOME NEW METHODS IN INTERIOR BALLISTICS*

BY ARTHUR GORDON WEBSTER

CLARK UNIVERSITY, WORCESTER, MASS.

Read before the Academy, April 26, 1920.

The principal problem of interior ballistics is, given a particular gun, a particular shot, and a particular kind of powder, to find, for a given load, the position and velocity of the shot, the mean pressure (and inci-

dentally temperature) of the gases in the gun, and the fraction of the

powder burned, all as functions of the time or of each other until the exit of the shot from the muzzle of the gun. In particular, we wish to

know the muzzle-velocity of the shot, the maximum pressure to which the gun will be exposed, and the portion of the bore which will be exposed to it. It is then the duty of the mechanical engineer to design a gun to

safely resist the pressure that may be expected. Or it is the inverse

problem of the ballistician, by experiments on the action of the powder in the gun, to find its properties and those of the gun. It is true that the

properties of the powder are more conveniently studied by means of

*Communication from the BAALISTIC INSTITUTE, CLARK UNIVERSITY, No. so.

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