"98 Transactions of the Royal Society of South Africa. 
(3) The most numerous and the most interesting of them are those that 
are irresolvable. As a typical example we may take 
la-fi^l \a^h^\ lacyb^^l 
la^Col kjCj^l (X0C4I 
l^i^al l^i^sl M4I y 
the irresolvability of which is implicated in the equality 
I \a^b.2\ k'lC.^ I 1^2^41 I = ki?>2C4.l la^b^dr^] + a^h^la^b^Cc^d^l. (I) 
One way of proving this may be put shortly as follows. In the given 
determinant 
the cof actor of l^gf^^l = ai^a-fi^c.^l, 
the cof actor of kgC^I = b-^\h-^aod^\, 
and the cofactor of la^b,^] = a^lb^eod^i + c-^^la-fi^d^l *, 
so that the determinant 
= \ajb^d^\ {cj\ac^b^\ — b-^laoC^} } 
+ 0-1 {1^4! ki&oCgl + \aobJ \b-^Ccyd.^\}. 
By increasing the first part of this by 
la-fi^d^i ■ cij • Ibc^c^l 
and diminishing the second by the same, the former becomes 
la^My! \a-^^b^c^\, 
and the latter becomes 
a-^^-^lb^d^l jfc^aiCgl — IM4I iWiC.J - Ib^c^^l Ifc^aAij 
which is readily seen to be equal to 
as desired. 
(4) We may note in passing that to the development thus obtained, 
\ajb.2C4} la^b^d^l + «;l^2I«i^2^3'^4I 
there is an alternative form, namely, 
[a^boC.^] \ajb2d^\ + a^bila^b^c^d^]. 
The equivalence of the two is established by noting that their difference 
la-Jy^c^l la^^fegcZgl - \a-JyoCr>^\ \a-J)od^\ + k^&gl la^b^c^^d^l 
is the extensional of 
c^d^ — 63^/4 + Ic^dJ 
which manifestly vanishes. 
* This identity used by Binet in 1812 : see/ Hist.' i, p. 89. 
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