DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
317 
art. 64 , by putting 2/,= and taking Z a function of x^, &c., y^, &c. and t, defined 
by th e equation Z = — W + 2^ (a? • y ,) , 
in which x \, ... x'^ are to be expressed in terms of y^, &c., x^, &c.*. 
Conversely, a system of the form ( 1 .) being given, it may be changed into a system 
of the form (80.) as follows: by means of the equations X(=.^, let 3/1, ... be ex- 
dyi 
pressed in terms of x\, &c., x^, &c. ; it follows from Theorem I. (art. 49 .) that we shall 
have 
rfW 
dx. 
I 
(a.) 
and 
dZ__ dW 
dxi dxi ’ ' 
where W is a function of &c., x[, &c. defined by equation 
W=-Z+X{x[y,), 
in which 3/1, ... are to be expressed in terms of x'l, &c., a:,, &c. The n equations 
dZ 
y\z=z—^ are then changed by (a.) and (b.) into the form ( 80 .). 
doc I 
On the Transformation of Coordinates. 
67 . It has been seen in the preceding article, that we can always change the system 
of 2 n equations of the first order of the form (I.), art. 64 , into a system of n equa- 
tions of the second order of the form ( 80 .). In this latter form the equations of 
dynamics naturally present themselves. 
Now in the case of the dynamical equations, x^,x^,...x„ are the independent 
coordinates of the system (the word coordinates being taken in its most general sense), 
and when the equations are to be changed into the form (I.), the additional variables 
3/1, ...3/„ are defined by the equations -^=3/.. In this case a transformation of coordi- 
nates, in the most general sense, consists in taking n new variables li, ••• L, con- 
nected with the original coordinates Xi,...x^ by n equations, which may also involve t 
explicitly. It is a well-known theorem, that the transformation of the equations ( 80 .) 
is effected merely by expressing W in terms of the new coordinates ... and their 
differential coefficients ...|^, instead of the old ; so that the new equations are 
d'w 
d^'J-d^, ■ 
( 81 .) 
the proof of this theorem does not depend upon the form of the function W ; and we 
know also (see arts. 18 & 66.), that whatever be the form of W, these new equa- 
tions may be again transformed to a system of the form (I.), by taking n additional 
* This theorem is a generalization of Sir W. R. Hamilton’s transformation of the Dynamical Equations, 
See Part I. art. 18. 
2 u 2 
