420 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
the theorem held in the case of n variables. The function being denoted 
by u, and being of the m th degree in the variables x v x 2 , , x n , Jacobi, 
like Hesse himself in his paper of 1847 (August), obtains the equations 
a? 1 w 11 + ^ 2 w 21 + +x n u n i = (?M-l)wf 
x Y u 12 + x 2 u 22 4- • • • + x n u n 2 = (?n - 1 )u 2 
and thence 
x 1 u ln +x 2 u 2n + ••• + x n u nn = (rn- l)u 7 
x { A = (m — 1 ) | U^! + U i2 u 2 + • • • + TJ in u n | 
Differentiating both sides of this with respect to x k , we have 
X^~ = (m- 1) | U il % 1 + U i2 % 2 + ' • • + ^in U kn | 
(“) 
+ 
(m — 1) | u x 
aUi 
dxj. 
SU,o 
+ u *d^ + 
+ u r 
au* 
dxi . 
\ 
/ i \ ( 31'hi 0U) 2 0U) n ) , m 
" (-- 1 )P l0 ^ + ^+ +u n ^\ (/?) 
if k be different from i. A second differentiation, but this time with 
respect to x t , gives 
0 2 A 
l 'dx 1 cxi 
/ IX f 3 2 D,t 
(??2 - 1) < U,—-^ 
7 ( 1 0J®C, 
+ W 
0 2 lL 
+ 
»0% dxfix^ 
0U, 
+ 
0 2 U, 
(m—1) | 2%^ +u l2 ^ 1 + 
I CXi. 
dXr. 
+ Un^ \ 
Wt ) 
I. 
1 
0^70^* 
+ u ln~ 
But on the supposition that l is different from i we have 
+ u l2 U i2 + • • • + u ln U in = 0, 
and therefore by differentiation with respect to x k 
0U a , d\J i2 , 
Uix-^-± + u l2 —^ + 
dx k &x k 
+ u l: 
% dxz. 
+ u kli^n + u m Ih 2 + • • * + U klnJ3 in = 0. 
Consequently by substitution 
M / i x f 3 2 U a , 
dxfixjc 7 1 dx t dx k 
... +w 9 -!^4 
n dxfix k f 
(m — 1) | UjfoU ft + 1l m V i2 + • * • +%ZrJJ'm |“ j 
0U, 2 
2 dx,dx-j 
(y) 
where l and k are each different from i. 
If now special values of x v x 2 , . . . x n make u v u 2 , . . ., u n all vanish, 
then, Jacobi says, we shall also have 
