1909-10.] Dr Muir on the Theory of Bigradients. 405 
Bruno, Faa di (1859). 
[Theorie Generale de l’Elimination. Par le Chevalier Francis Faa di 
Bruno. . . . x + 224 pp. Paris.] 
In his section (pp. 32-40) dealing with the dialytic eliminant, Bruno, 
besides reprinting R 33 , R 4 4 , gives the full expansion of the resultant of 
ax 3 + Sbx^y + Sexy 2 + dy 3 — 0 | 
bx 3 + Scx 2 y + 3 dxy 2 + ey 3 — 0 / 
and of the resultant of 
ax 4 + 4:bx d y + Qcx 2 y 2 + kdxy 3 + e?/ 4 = 0 ^ 
bx A + icx 3 y + Qdx 2 y 2 + 4 exy 3 +fy 4: = 0 j 
— that is to say, the full expansion of the discriminant of the equation 
ax 4 4- 4 bx z + Qcx 2 + 4 dx + e = 0 
and of the discriminant (with at least seven mistakes) of 
ax^ + 5foc 4 + 10 c£ 3 + 10^ 2 + 5ex+f = 0. 
In the next section (pp. 40-46) he seeks to improve on what we have 
called Cayley’s “ chain of squares ” by combining the last square with the 
first, the second from the end with the second from the beginning, and so 
on. For example, his expression for R 33 , that is to say, for 
(a 3 s 3 - <i 3 j? 3 ) + ( - a 2 brs 2 + cd 2 p 2 q) + {2( - a 2 cqs 2 + bd 2 p 2 r) + . . . } + ... 
is 
jps 2 qrs r 3 prs q 2 s qr 2 
a marked improvement on which would be 
s 3 rs 2 
p 3 p 2 q 
+ . . . . 
the second term in each binomial being derived from the first term by the 
ohange of a, b , c, d, p, q, r, s, into d, c, b, a, s, r, q, p , — that is to say, by the 
interchange 
7a b p q\ 
| d c s r) . 
Further, he improves upon Cayley’s squares by so transposing their rows, 
+ 1 
+ 
a 2 b 
cd 2 
