MAXIMA AND MINIMA 
183 
The condition here takes on the form : 
where i and j, independently of each other, run through the values 
from 1 to n, and a,-, = a, 0 equations (8) assume the form 
«11*1 + ••• + <*i „*„ d* ^- x i — 0, 
<*21*1 d" 4" a 2n X n d* ^-*2 = 
(14) 
«„1*1 H h <*„„*„ + Xx n = 0. 
Hence A. must be a root of the equation 
«n -f- X ctj2 .... «i„ 
«.¿I «22 d" A. ... «2„ 
«11 + X «12 • 
«21 «22 d" ^ 
= 0. 
(15) 
a nn + ^ 
a»i a „2 
6. Show that if, in the preceding question, the function u has a 
maximum or a minimum in the point (0, 0, ••*, 0, x' H ), where x' n > 0, 
then F contains only a simple term involving x n , namely a nn x 2 n . 
7. A “ rotation ” of space of m dimensions is given by the formula 
k = 1, •••, m, 
*! = <** 1*1 + — +%m*. 
where 
provided the determinant of the transformation (which = ± 1) has 
the value + 1. 
Show that, by a succession of rotations (which can, of course, be 
compounded into a single rotation) the form F of Question 6 can be 
carried into a form in which only the terms in x\ are present. 
8. Find the points of the circle 
X 2 q. y2 -q_ s: 2 == 1, ax + by + cz = 0, 
in which the function 
w = Ax 2 + Bf + Cz 2 + 2 Dyz + 2 Ezx + 2 Fxy 
attains its greatest and its least values. Treat first the case: 
I) = E = F = 0.