682 
Proceedings of the Royal Society of Edinburgh. [Sess. 
The first matter of real interest is reached on p. 11, where the following 
theorem is given: “ Soient s la resultante de Vordre i formee avec les 
termes du tableau 
a 0 a x ... 
b 0 b x • • • l J i - 1 
l 0 l x ... l { _ i , 
et s" la resultante de Vordre i — 1 formee avec les termes compris dams le 
tableau 
S( ± a Q b x ) S( ± a 0 & 2 ) .... S(± a 0 6<_i) 
S( ± a 0 c x ) S( ± a 0 c 2 ) .... S( ± a^) 
S( ± a 0 h) S( ± a 0 J 2 ) .... S( ± V*-i) . 
La resultante s" sem egale a s, au facteur pres a* , en sorte quon aura 
s'^a^ 2 s” 
This is one form of the theorem afterwards well known as effecting the 
transformation of any determinant into one of the next lower order. It 
may be viewed as a case of Hermite’s result of the year 1849. 
On p. 17 particular cases cease to be considered, and the multiplication 
of an array of i rows and 2 i columns by a similar array is taken up, with a 
result in accordance with that arrived at by Binet and Cauchy in 1812. 
From this result, by specialisation, the ordinary multiplication-theorem is 
then deduced, and with it (Ohio’s “ theoreme ix. ”) the first part of the memoir 
closes. 
The second part, which begins on p. 23, concerns the solving of a set of 
2 n equations of a type which will be sufficiently specified by giving the set 
where n = 3, namely, 
x + y + z — d x 
x£ + yr) +z£ = d 2 
x^ + yf + zt, 2 = dg , 
x^ + yf + z^ 5 = J Q . 
The connection of this with what precedes consists in the fact, arrived at 
by Sylvester in his solution of the problem of the canonisation of the quintic, 
that £ tj, f are then the roots of the equation in co 
d 2 — o od x 
d g (l ct .) 
d 4 — iod a 
d 3 — lode, 
d x lod^} 
dr o — lod x 
d 4 iod^ 
d 5 - iod 4 
d 6 — iod 5 
0 . 
