1901 - 2 .] Dr Muir on the Theory of Orthogonants. 
265 
“les expressions 
*^i » Vi ’ % 5 • • • • 
x 2 ’ V2 ’ % ’ • ‘ ' ' 
“seront, aux signes pres, completement determinees . . . . , 
“ a moins que des racines de l’equation S = 0 ne verifient en 
“ meme temps la formule 
P« = 0 
The next step is to prove that the roots of the equation S = 0 are 
all real so long as the coefficients of the quadratic s are real. If 
ihe contrary he supposed, viz., that one of the roots s p is of the 
form \ + fi J - 1 , this will of course entail the existence of another 
s q of the form A. - /x J — 1 . Also, X p being the same function 
of s p , that X s is of s q , it will follow that X p and X q will be of the 
form 
M + NV^I , M-N^l 
and therefore that 
X,X 2 = M? + jS t * 
This means that X p X q will be positive or zero, and similar reason- 
ing would prove the same regarding Y p Y q , Z p Z q , . . . None of 
them, however, can be positive ; for since 
% + VpVq +•••* = 0 , 
it follows from the values obtained for x p , x q , . . . , that 
XpXg + Y p Y q + • • • = 0 . 
And since they are all zero, and each the sum of two squares, 
we are forced to the conclusion that 
= Xg = 0, 
Y„ I Yg = 0, 
Zp = Z q = 0 , 
which is the same as to say that the roots x pi x p satisfy the 
equations 
0 
