508 Proceedings of Royal Society of Ediiiburgh. 
which comes immediately after (C), and is derived from it by ex- 
tending the factors in which af>^ does not occur. Since by defini- 
tion 
\af)^cfl^ = d^af.2p^ - -l- dfoL-pgi^ ~ , 
and \af 2 ^fl-^ ~ drfif 2 ^^ ~ ^3l^i^2^5l ~ ? 
it follows that 
l^l^2^5ll'*1^2^3^4l “ l^l^2^4lk-h^^2^3^5! 
= I d^gf)2^^ ~ ^5l®i^2^4| } l^l^2^3l 
+ { |<^1^2^5ll^l^3^4l “ l%^-^2^4il^l^3^5l }■ 
I 
- ■{ - \a^h^e^^ajj^c^\ }di . J 
But the cofactor here of d^ is by (C) equal to 
— \af>^c^af>2^^. } 
and the cofactor of d-^^ 
= \aAhWV^i\ - I«2^iC4!1“2^5! - 
and therefore by (C) 
= — \af>iC^af>^c^ , 
= • 
Making these substitutions, we have 
l^l^2*^5ll^l^2^3'^^4l “ l'^l^^2^4ll^l^2'"3^5 
= ~ l^b^2^3l { ^51^1^2^41 — ^4kb^2^5l "f ^^21^1^4^51 ~ ^ll^2^4^5l}’ 
= - \af2(^fgf)2P^d ^\ , 
as was to be shown. 
The next three cases are 
- i«lV4^6ll«1^2^3^5i + |«1^2^3'^^6ll^l^2^4f^5i = ^ (^) 
lcq&2%^4l!^1^2^3^5^6i “■ I^1^2^"3^5lk'b^^2^3^4^6! d' l^l^2^3^6il^l^2^3^^4^5! ~ ^ (^) 
!(ll&2^3^4^7ll^l^2^3^^5^6i ~ l^i^2^3^5%H^1^2%^4^6i d" 1^1^2^3'^6*^7ii^l^^2^3^^4^5l ~ ^ (^)* 
When the factors of each product are of the same order, as in 
(C), (E), (G) the identity is, in modern phraseology, an “ exten- 
sional ” of (A) ; that is to say, there is a part common to every 
factor of the identity, e.g., cq in (C), a ^&2 ^i^ 2^3 (^)> 
this common part being deleted, the result is simply the identity 
