DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
83 
the expression [/, g], or 
f dtp d\ J/ d<p 
l \dyi dxi dxi dyA ’ 
is constant ; i. e. it becomes a function of the arbitrary constants only, if for x { , &c., 
&c. be substituted their values in terms of the constants and t. 
In the case in which (16.) and (17.) represent the dynamical equations, this is 
identical with the remarkable theorem discovered by Poisson. We shall have occa- 
sion to return to it presently. 
10. If we treat the equations (21.) of the last article exactly in the same way as 
we have treated (22.), putting u , v for any two functions whatever of the 2 n variables 
we find 
Xi, x 2 , ... x n ,y x ,y 2 , ...y 
dju, v) 
‘d(bi, flj) i d(x i ,y i )'‘ 
nl 
and comparing this with the theorem (25.) of the last article, we see that both may 
be included in the following general enunciation: — 
If u, v be either (1) any two functions whatever of the 2 n constants a x , &c., b x , &c., 
or (2) any two functions whatever of the 2 n variables x x , &c\, y x , &c. (not containing 
t explicitly), then 
or 
J du dv du dv du dv du dv 
\dyi dxi dxi dyi db { da x da x db t 
= 0 , 
^ f d ( u > v ) 
M %*, ®i) 
1 
r <Hb„ a ,) J- U 
(26.) 
(When m, v represent functions of the constants, the differential coefficients in the 
first term are taken on Hyp. II. ; and, when functions of the variables, those in the 
second term on Hyp. I. (art. 5.)). 
This property depends, as will be seen, solely on the relations (5.), (11.), arts. 2, 5, 
which are the only assumptions that have been made in deducing all the preceding 
propositions. 
11. There are similar theorems in which the summation refers to the numerators of 
the differential coefficients ; but as these are less remarkable, and moreover are 
deducible immediately from the equation (20.), art. 8, 1 shall omit them. 
12. Theorem . — I proceed now to establish a theorem which may be considered as 
the converse of that expressed by (23.), art. 9. 
Let x x , x 2 , ... x n , y x , y 2 , ... y n be 2 n variables, concerning which no supposition what- 
ever is made, except that they are connected by n equations 
a x — (p x {x x , x 2 , ... x n , y x , y 2 , ... y n ) 
a 2 — l p 2 ^x x , x 2i ... x M y x , y 2 > ... yn) 
(a.) 
a x , a 2) ... a n being n constants. The functions on the right may involve explicitly any 
m 2 
