Dr T. Muir on the Theory of Determinants. 
511 
~ \ \ ^ll^3^^4l 
= 0, 
— I 1 
— h.feyd^ + j- ^ 
+ } 
+ 5g|C^C?2l I" ^ ) 
for the terms in the expanded form destroy each other in pairs. 
The derived identities are obtained exactly in the manner 
followed by Bezout in 1779 (see pp. 457-8, Vol. XIV.). The funda- 
mental identity is taken, say in the form 
f ^5l'^1^2^3^4/5l - “ ^sKb^2'"^3^4/5l 
+ 551^1^20^364/51 - %i5iC26/364/5| = 0, 
and another instance is put alongside of it, in which the same 
letters and suffixes are involved, say 
/ll«l¥3^4^5l - 61100 / 2 ^ 3 ^ 4 / 5 ! + 051 I 00 / 2 C 364 / 5 I - 6i!o0/26Z3e4/5l 
+ t)-^\aiC2dQe^f 5] ~ 
One of the constituent determinants, say the last, 15462 ( 7364 / 5 ! is 
then eliminated by equalisation of coefficients and subtraction, the 
esult being 
l«i/5l • \aAh<^ier} - l«lC5ll«lV3<^4/5l + 
+ - \aAP-i<^2'^-i<^Jz\ = 0 (C'") 
In the next place, two additional instances of this derived identity 
are taken along with it, the first differing from it in having a 2 
instead of a 5 in all the first factors, and the second in having a 2 
instead of a 1 ; viz., 
|6^i/ 2!!%^263^465! “ !o^l62l!o^/263^4/5l + !f0’ A!!6^/26364/5! 
4- !o0iC2!|004526?364/5l - |0Oi52!io0i626?364/5| = 0, 
and 
\a-2frp,Avh<^z\ - + A'^'MA:Pz'^Ji\ 
