1901- .] Dr Muir on the Theory of Orthogonants. 
26 $ 
Theoreme I . — Quel que soit le nombre n de variables x, y, w 
z , . . . V equation 
S = 0 
et les equations de meme forme 
R = 0, Q = 0, .... 
auront toutes leurs racines reelles. De 'plus, si I’on nomme 
s', s", s'", . . . , s (n - 1} 
les racines de V equation 
R = 0 
rangees par ordre de grandeur , les racines reelles de V equation 
S = 0 seront respectivement comprises entre les limites 
oo , s , s , s 
00 . 
Considerable space (pp. 188-192) is next given to extending 
this theorem to the case where several values of s satisfy at the 
same time two consecutive equations of the series S = 0 , R = 0 , 
P = 0, .... 
Then follows a series of noteworthy deductions, which bring us 
round to the solution of a general problem of a quite different, 
character, viz., the problem of transformation which we have seen 
Jacobi attacking in detail. Denoting, as before, the extreme 
values, all different, of the quadratic function A^ 2 + A yiJ y 2 + • • • 
+ 2A xy xy + • • by s ± , s 2 , ... , s n , and by x r , y r , z r , ... the 
values of the independent variables which give rise to s r , we know 
that we have 
(A» 
-*iK 
+ 
A „/j 1 
+ 
+ • 
• = 0 
+ 
{^-yy 
~ s i)Vi 
+ 
Ay^Z 1 
+ • 
. = 0 
+ 
Kvi 
+ 
(A« 
- s iK 
+ • 
• k o 
xf 
+ 
Vi 
+ 
^ 2 
+ 
• = i 
(A« 
- S 2 )x 2 
+ 
A xy y 2 
+ 
A X7 z 2 
+ • 
• = 0 
A xy X 2 
+ 
(K 
~ hYdz 
+ 
A yz z 2 
+ • 
• = 0 
A xz x 2 
+ 
A yz y 2 
+ 
{K z 
- hK 
+ • 
• = 0 
ry » 2 
+ 
v-2 
+ 
z 2 2 
+ • 
. = 1 
) 
V 
