DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
87 
but 
d(7i) rfZ dy x <TL dy <2 
da.i dy l dai'dy 2 da t ' 
(since (Z) is derived from Z by introducing the values of y x , ... y n , in terms of <r„ See., 
a 19 ... a n ), hence the second member of the preceding equation vanishes, and we have 
dX\' 
daj 
dX. 
so that is constant, and we may write 
dX , 
(III.) 
and bi is an independent arbitrary constant, as it is easy to prove ; it is however 
unnecessary to do so here, because we have in fact already proved it in showing that 
the elimination of ... a n , b lt ... b n , from the system of equations (II.), (III.), leads 
to the differential equations (I.) (see art. 6.). The n equations (III.) give therefore 
the remaining n integrals of the system (I.), of which (II.) and (III.) together are the 
complete solution. 
The system of equations (II.), (HI.) being the same as that discussed in the pre- 
ceding articles, all the conclusions there obtained will continue to subsist. 
15. Suppose the expression for Z (see the last article) in terms of the variables is 
Z =/(*!, ^ 2 , ... x n , y x , y 2 , ...y n ,t), 
d~X. 
Z is changed into (Z) by the substitution of - for y 1} &c. 
dX 
and since is (identi- 
cally) = — (Z), the equation 
x 0 
X n 
dX 
dx x 
(X.) 
is a partial differential equation satisfied by the function X. 
We have thus arrived, by an inverse route, at the point from which Sir W. Hamil- 
ton’s theory, as improved by Jacobi, sets out. 
Jacobi, namely, has shown (by a demonstration immediately applying only to a 
particular form of the equation (X.), but easily extended), that if X be any “ com- 
plete ” solution of the equation (X.), that is, a solution involving (besides the constant 
which may be merely added to X) n arbitrary constants a 1} a 2 , ... a n , in such a way 
that they cannot be all eliminated from the w+1 equations obtained by differentiating 
X with respect to x x , ... x n , t , without employing all those equations, then X possesses 
the properties of Sir W. Hamilton’s “Principal Function,” or in other words, gives 
all the integrals of the system (I.) by means of the system (II.), (III.). It will be 
desirable briefly to indicate the mode in which this demonstration may be made to 
apply to the general form (X.). 
dX 
Assuming that a complete solution X, of that equation, is given, put then 
