262 Proceedings of Royal Society of Edinburgh. [sess. 
and therefore by the fundamental theorem regarding the 
differentiation of homogeneous functions and by the above- 
mentioned condition 
\= 2s . 
He thus obtains the set of equations 
or 
^ = sy ’ te = S2 ’ 
! 0Z 
(Rxx - S ) X + RxyV + Rxz Z + • * • = 0 \ 
A y& + (A yy -s)y + A yz z + ••• = ()! 
A zx x + A zy y + (A zz -s)z + • • • = 0 [ 
and therefore concludes that, on eliminating x , y , z , . . . from the 
set, the resulting equations in s , 
S = 0 
say, has for its roots the maxima and minima values of s. The 
third chapter of the Cours dJ Analyse is then referred to and taken 
as warrant that 
“S sera une fonction alternee des quantites comprises dans 
le Tableau 
A xx s A x y A xz .... 
A A q A 
■^xy -“ 2 / 2 / ° .... 
A xz A yz A sz -s .... 
and the developments of the function are given for the cases 
n = 2 , n = 3 , n = 4 exactly in the form adopted by Jacobi. 
The question of the particular values of the variables x , y , z , . .. * 
which correspond and give rise to each of the n extreme values of 
s is next taken up, the equations for the determination of them 
being clearly the set from which the equation S = 0 was obtained 
(a set, be it remarked, which of itself can only give the ratio of 
any two) and the additional equation 
x 2 + y 2 + z 2 + .... = 1 . 
A series of identities connecting these n 2 values is however first 
obtained. Denoting by x r , y r , z r , . . . the values of x , y , z , . . 
