302 
PROFESSOR DONKIN ON THE 
(i being different fromj); and obviously in all cases 
[;?, = ;?], and \_p,p]=0 (art. 9.). 
Theorem V. — If w, v be either (1) any two functions whatever of the 2n constants 
ai, &c., &c., or (2) any two functions whatever of the 2n variables &c., ?/„ &c. 
(which may in either case also contain t explicitly)*, then 
y j du dv 
\dyi dxi 
du dv'] ^jdu dv du dv'\ 
dxi diji] idui dbi dbi dai\ 
(59.) 
(When u, v represent functions of the constants, the differential coefficients in the 
first member of this equation refer to Hijp. II.; and, when functions of the variables, 
those in the second member refer to Hyp. I.) (art. 10.). 
Theorem VI. — Let x^, ... x^, y^, ... y^ be 2n variables concerning which no supposi- 
tion is made except that they are connected by n equations of the form 
'^25 3^25 ••• 2 / n ) (^•) 
(where the functions on the right are only subject to the condition that the w equa- 
tions (a.) shall be algebraically sufficient to determine 3/1, ...?/„ in terms of Xi,...x„, n,, 
&c,, and may contain explicitly any other quantities besides x^, &c., y^, &c.). 
Then, if by means of the equations (a.) the n variables 3/1, 3/2, ... y^ be expressed as 
„ n r> • 1 1 I n{n—\) ,, . 
functions of .Tj, Xg, &c., &c. ; in order that the — ^ — conditions 
dyi_dji 
dxj dXi 
may subsist identically, it is necessary and sufficient that each of the 
n{n — 1 ) 
2 
expres- 
sions [«„ aj vanish identically. 
Theorem VII. — Let Z be any function whatever of 2n variables Xy ... x„, y^ ...3/„, and 
t. If of the system of 2n simultaneous differential equations of the first order 
dxi 
(I-) 
there be given n integrals involving n arbitrary constants a^, ... so that each of 
these constants may be expressed as a function of the variables x,, &c., 3/1, &c. (with 
or without t ) ; then If the 
n(n-l) 
conditions [a;, aj] =0 subsist identically , the remaining 
n integrals may be found, as follows. By means of the n given integrals let the n 
variables 3/1 ...3/^ be expressed in terms of x,, &c., a^, &c. ; and let (Z) be what Z 
becomes when y^ ...y^ are thus expressed. These values of 3/1, 3/2 ...3/„and — (Z), will 
be the partial differential coefficients with respect to Xy, x^, ... x„ and t, of one and the 
same function ; call this function X, then, since its partial differential coefficients are 
* It was inadvertently stated in art. 10, that u, v must not contain t explicitly. But it is evident that no 
such limitation is implied in the demonstration of the theorem. The preceding theorem is obviously a particular 
case of this ; namely, the case in which u=aj, v=bj. 
