DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
73 
and attribute a corresponding - meaning to 
where d l} d 2 , ... d n are symbols denoting n distinct and independent sets of variations, 
so that 
d r y t 
dX-y UO, 2 
then it follows from well-known properties of determinants (as M. Bertrand has 
shown) that the complete functional determinant formed with the n 2 differential 
coefficients 
%i, _ dya 3 
dxy dx 2 ' dx x dx 2 
is equal to the quotient of the two determinants which I propose to denote by 
y 2 , y 3 , ••• y n ), d(x u x 2 , x 3 , ... a?,), 
and moreover that the partial functional determinant formed with the m 2 terms 
dy t dyi 
dx p dx q 
d Ji, 
* *’ dx p dx q 
7i, &C. 
is equal to the quotient of the two partial determinants 
d(yi, y j3 y k , . . .), d(x pf x q , x r , ...), 
the differentials of y t , &c. being taken on the hypothesis that all the differentials of 
the x- variables are =0, except those of the set x p , x q , x r , Thus the expression 
(D.) is a real fraction, provided its numerator and denominator be interpreted in a 
manner exactly analogous to that in which the numerator and denominator of an 
ordinary total or partial differential coefficient are interpreted.] 
This being premised, let u x , u 2 , \bem functions of any or all of the functions 
y 1} y 2 , ... y n {m being supposed not greater than n ), so that m,, u 2 , &c. are functions 
of x lf x 2 , &c. through y 1} y 2 , See. 
Let any selected sets of m indices out of the series 1,2, ... n, be denoted, for greater 
clearness, by a„ « 2 , ... a m ; (3„ (3 2 , ... (3 m ,8cc. Then the general theorem analogous to 
7 dui J dill 7 | Q 
*‘=^+^+ &c - 
may be expressed as follows : — 
j d(u ai , M a-2 , 
• •> U «m) i 
1 %*»,, Vh, 
"> ypmj 
d (y^y^ •••>#&»)} 
(the summation on the second side referring only to the indices (3, and extending to 
every combination of m out of the n numbers 1, 2, ... n ). 
In like manner, the theorem analogous to 
dui 
_i_ dyd dy x dui dye , 
dxj dy x dxj'dy 2 dxj 
IS 
d(u, 
a,, u a 2 , 
I y J d(u a lS u a , 2 , . . . d(y Pi , yp 2 , 
yp J] 
d{x. 
viJ x yi’ 
••• x yJ 
\.d(yp,, yp& ••• yp m ) d{x Vl , x y „, ... x 7m )j 
MDCCCLTV. 
