648 
PHYSICS: A. G. WEBSTER 
Proc. N. a. S. 
The above semin variants may be converted into semin variants 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 iriki by Uiki, ir'iki by Viki and 
TTiki by Wiki. 
A comparison of the seminvariants I^'^^^ of (A) with the corresponding 
seminvariants^ for the case n = 2 shows the former to be independent. 
Moreover, the functional determinant of li''^^ with respect to Uiki for 
«ach value of / = 0, 1, 2. . .,w - 3 shows that I^''^^ are independent 
among themselves and of the seminvariants 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^'''^^ and We have therefore the following 
theorem : 
All seminvariants of (A) are functions of I^^^^^ {r = .,n — 1; r + 5 < 
n) t = l,2,3',t < s), It''^ (f = 0, l..,n - I', r ^- s < n\ t = 1, 2; 
t <s;l = 0,1,. .m - 3), and of the derivatives ofl^''^^ {t = 1,2) andli''^\ 
^ Wilczynski, 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 Balwstic Institute, Clark University, No. 10. 
