-A (i s + v 3 + Ç 3 ) 2 
+ (12PA + 4IB) (p + v 3 + ?) ÇyÇ 
+ (181A+(1+8P)B) p?? 2 p 
+ ((4£ 2 + 8P) A + 4PB) ( V 3 Ç 3 + pp + pp). 
The coefficient of pp + pp + pp on the right-hand side will vanish if (1 + 21 s ) A + P B = 0, 
or, what is the same thing, if A = l 2 , B = — (1 + 2£ 3 ) ; and substituting these values, we 
obtain 
{P (*> + y z + ¿3) _ (i + 2i 3 ) 4- (p - p) (p - p) (p - v s ) 
= ~ P (I s + p + p) 
+ (- U + 4Z 4 ) (p + p + p) p? Ç 
+ (- 1 + 81 s - 1QP) ppp, 
or, what is the same thing, 
P (x 3 4- y 3 + z 3 ) — (1 + 2Z 3 ) xyz = — (p — p) (p — p) (P — p) 
x {- Z (p + p + p) + (- 1 + 4Z 3 ) p?p 2 . 
Hence the left-hand side vanishes in virtue of the relation between p y, p or we have 
P (x 3 + y 3 + z 3 ) — (1 + 2P) xyz = 0, 
which proves the theorem. 
36. Suppose that (X, Y, Z) are the coordinates of a point of the Hessian, and 
let (P, Q, R) be the coordinates of the point in which the tangent to the Hessian 
at the point (X, F, Z) again meets the Hessian, or, what is the same thing, the