285 
1 909-1 0.J Dr Muir on the Theory of Orthogonants. 
in 0 2 with positive coefficients, and that when n is even it is already of the 
latter form. All values of 0 2 thus obtainable must be negative, and conse- 
quently all the values of 0 save the value 0 must be imaginary and must 
occur in pairs whose sum is zero. But as 
and 
= 1 
A x-l 
^(l+a)A x+l 
x 
1+2 
it is clear that for every pair of values of 0 that differ only in sign there 
must be a pair of values of x that are reciprocals. The theorem reached 
by Brioschi we may thus enunciate for ourselves as follows : — The roots of 
the equation 
<°I] X w 12 
• • W ln 
W 21 0 22 ~ X ' 
• • w 2 n 
<%L W /(2 
. . 0) nn 
where | <b u co 22 . . . o) nn | is Cayley’s orthogonant , are arrangeable in pairs of 
reciprocal imaginaries, save when n is odd , in which case there is the 
single real root 1. 
When instead of the co’s we take the coefficients of the substitution 
which transforms a general quadric into itself, the words “ reciprocal 
imaginaries ” need to be changed into “ reciprocals.” This generalisation 
Brioschi published a month or two sooner (see Annali di sci. mat. efis., v. 
pp. 201 - 206 ). 
Bruno, Faa de (1854, September). 
[Note sur un theoreme de M. Brioschi. Journ. (de Liouville) de Math., 
xix. p. 304.] 
On multiplying both sides of Brioschi’s equation (1854, August) by 
| co n o) 22 . . . co nn | and dividing by ( — x) n an equation is obtained which differs 
from the original simply in having x~ x for x. The portion of the theorem 
which concerns “ reciprocity ” Faa de Bruno thus readily establishes. 
