DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 315 
(see Theorem III.). And if p, q be any two functions of the variables (with or with- 
out f), then 
\dyidxi dxidiji) ^\d-rii d^i d^i driij 
where/? and q in the first member are supposed to be expressed in terms of j?,, &c., 
//„ &c., and, in the second, in terms of &c., ni, &c. In other words, the value of 
[/?, q'] is the same, whether it be obtained from the expressions for p, q in terms of 
the original variables, or by an analogous process from their expressions in terms 
of the new. 
Particular cases of (77») are the relations 
[li, = - 1 , !•] = [% yii] = [li, y = 0 (78.) 
(See Theorems IV., V.) 
64. The relations (74.), (76.), (77-)? (78.) of the last article depend solely upon the 
form of the equations (73.), art. 62, which connect the new variables with the old ; 
and are independent of any supposition as to the equations which may determine 
either set of variables as functions of t. Let us now, however, introduce the suppo- 
sition that the original variables Xj, ... x^, ?/,, are determined as functions of t by 
the system of differential equations. 
X 
dZ 
dyl dxi 
(I-) 
The relations just established enable us immediately to transform this system into 
another involving the new variables instead of the old ; for we have 
now 
(*ee(74.),a, t.63); 
and if in the remaining term we substitute for x'^, y] their values from (I.), and for 
— their values it becomes 
axj dyj drii dt^ 
/JZ dXjj^dZ 
dr^i dyj driJ‘ 
dZ 
which is equivalent to if Z be supposed expressed in terms of the new variables. 
We have then 
and, exactly in the same way. 
^,_^,dZ 
d^i d^i 
This result may be stated in the form of the following Theorem VIII*. If the system 
* This theorem, in its general form, is, to the best of my knowledge, new. But that case of it in which P 
does not contain t explicitly has already been proved in a different way by M. Desboves, who has, by means 
MDCCCLV. 2 U 
