90 
PROFESSOR DONKIN ON THE 
similar transformation to the equations (T.), in the case in which no limitation is im- 
posed upon the form of the function T, as I shall now proceed to show. 
18. Putting T-fU=W, we shall have (since U does not contain x\, Sec.) 
/dW\' dW 
\ dxi ) dxi 
(W.) 
dW 
Let — j—y ii then if we take 
Z= — (W) + (x\)y l + (x' a )y 3 +... + (x' n )y n (V.) 
(where, in the terms enclosed in brackets, x\, x' 2 , &c. are to be expressed in terms of 
y x , Sec., x i, x 2 , See.) , we shall have, by the theorems of the former articles (see equa- 
tions ( 6 .), ( 8 .), (9.) of arts. 2 and 3, putting x' t instead of x t and x t instead of p, in those 
equations), 
: dZ , N. 
#•■=— 3 (a.) 
d Ji 
and 
so that the equation (W.) becomes 
dW_ 
dxi 
'dZ 
dxi 
_ dZ 
Vi dx / 
m 
and (a.), ((3.) are of the form in question* ((I.), art. 14.). Thus, so far as the appli- 
cation of any methods of integration, founded upon the preceding principles, and the 
theories of Sir W. Hamilton and Jacobi, to the system (T.), art. 16, is concerned, 
there is no restriction to the form of the function T. This extension is probably at 
present of no practical importance, but may perhaps be thought of some interest in a 
purely analytical point of view. 
19. Returning now to the suppositions and conclusions of art. 14, let us further 
suppose that Z does not contain t explicitly, so that 
Z'=2, 
dZ 1 . dZ 1 
dx t Xi+ ^' 
=0 
by virtue of the system (I.) ; in this case 
Z =h 
(h.) 
is one of the integrals of the system, and if we suppose this to be one of the n given 
integrals from which the principal function X is to be found, so that 
h , ^13 ^23 ••• &n - 1 
are now the n arbitrary constants, and the conditions 
[«*,«,•]= 0, [A, aj =0 
subsist, it is plain that we shall have 
(Z )=h, 
* In the case in which T is homogeneous and of the second degree, in x' v x 2 , 
expression for Z reduces itself to 2(T)— W, or (T) — U. 
x„, it is obvious that the 
