516 Mr. A. Cayley's Demonstration of a Theorem 



The identity to be established is 



(?i;2 ^ ^2 _^ ^2 ^ . . ,) („,^2 + ff^ -b;"..) 



=v+«/+v+ ••• 



where the 2" quantities iv, a, b, c . . . and the 2" quantities 

 w,, a I, bi, Cf . . . are given quantities in terms of which the 2" quan- 

 tities iVii, 0^, bii, Cii . . . have to be determined. 



Without attaching any meaning whatever to the symbols 

 Aq, Jq, Cq . . . I write down the expressions 



io + aa^ + bb^ + cc^ ..., io, + a,a^ + b,b^ + c,c^ ..., 



and I multiply as if a^, b^, c^ . . . really existed, taking care to 

 multiply without making any transposition in the order inter se 

 of two symbols a^, 6^ combined in the way of multiplication. 

 This gives a quasi-product 



WW, + {aw, + a,tv)a^ + {bw, + b,w)b^ + . . . 



+ aa,a^^ + bb,b^^+ ... 



Suppose, now, that a quasi-equation, such as 



means that in the expression of the quasi-product 



*o^o' ^o"o> f'oK <^oh> «o^o> *o«o 



are to be replaced by 



«c» *o' <^o' -"cy -K> -(^o> 

 and that a quasi-equation, such as u^b^c^^—, means that in the 

 expression of the quasi-product 



*o^o> ^o«o' «o*o' ^o^o' «o^o» *o«o 



are to be replaced by 



-"o> -^o> -^o' Co> ^'o> ^o- 



It is in the first place clear that the quasi-equation a^b^c^^ + 

 may be written in any one of the six forms 



%<^oh=-> ^oV'o=-^ ^o«o^o=-; 



and so for the quasi-equation aj)^c^= —. This being pre- 

 mised, if we form a system of quasi-equations, such as 



«oVo=±^ «o^o«o=±&C., 



where the system of triplets contains each duad once, and once 

 only, and the arbitrary signs are chosen at pleasure ; if, more- 

 over, in the expression of the quasi-product we replace a^, b^, . . . 

 each by — 1, it is clear that the quasi-product will assume the form 



