276 
Proceedings of the Royal Society of Edinburgh. [Sess. 
and 
+ ^1^2 + ^31^3 + hl^4 = ^1 
he does not seek to ascertain therefrom the linear substitutions connecting 
the os’s with the g s, but bringing forward the coefficients of these substitu- 
tions as found by Cayley, namely, 
^ii _ | 2 Li 2 ^13 2L ]4 \ 
A A A A 1 
he affirms that by using as multipliers, along with the first set of equations, 
the elements of the various columns of this array in succession we shall 
have 
1 A A 
and that by using along with the second set of equations the elements of 
the various rows of the array in succession we shall have 
+ 
At the close of a preceding section (§ 6) he devotes three pages (pp. 35- 
37) to an investigation of the conditions under which Lagrange’s deter- 
minantal equation shall have all its roots positive. The result is not so 
interesting in connection with our present subject as a theorem made use of 
in the process of attaining it, namely : — If we have given the set of equations 
& n x x + a 12 ^ 2 + . . 
+ 
jo 
II 
j? 
a 21 ^! + a 22 a? 2 + . . 
■ • &2nP^n — @Xo 
a n i*i + a n2 * 2 + . . 
■ • ^nvP^n ~ _ 
where a rs = a sr , and if we put A for | a n a 22 . . . a BB | , then 
An*! + A 21 a? 2 + . . . +A nl x n = -x x 
A 12 *i + A 22 £ 2 + . . . + A n <fc n — — * 2 
V >• 
A 1b #i + A 2n £ 2 + . . . +A nn x n = -x 
v 
No proof of this is given, but one is readily got by using the elements of 
