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VII. On the Theonj of Groups, as depending on the Symbolic 

 Equation 6"=!. By A. Cayley, Esq.* 



LET ^ be a symbol of operation, which may, if we please, 

 have for its operand, not a single quantity x, but a system 

 (a-, y . . . ), so that 



e{x,y...) = {a;',y'...). 



Where x', y' . . are any functions whatever oi x, y . . . , it is not 

 even necessary that x', y' . . should be the same in number with 

 X, y . . . In particular x', y', &c. may repi'esent a permutation 

 of X, y, &c. Q is in this case what is termed a substitution ; 

 and if, instead of a set x,y . , , the operand is a single quantity x, 

 so that 6x-=x'=fx, 9 is an ordinary functional symbol. It is 

 not necessary (even if this could be done) to attach any meaning 

 to a symbol such as 6 + <p, or to the symbol 0, nor consequently 

 to an equation such as ^ = 0, or d±(f) = 0; but the symbol 1 

 will naturally denote an operation which (either generally or in 

 regard to the particular operand) leaves the operand unaltered, 

 and the equation d = <p will denote that the opei'ation 6 is (either 

 generally or in regard to the particular operand) equivalent to (p, 

 and of course ^ = 1 will in like manner denote the equivalence 

 of the operation 6 to the operation 1 . A symbol 6^ denotes the 

 compound operation, the performance of which is equivalent to 

 the performance, first of the operation (j), and then of the opera- 

 tion 6 ; ^(^ is of course in general different from (})6. But the 

 sjmibols 6, (j> . . are in general such that 6 . ^t^=^0 . ■^j &c., so 

 that ^0%, ^</>%<», &c. have a definite signification independent of 

 the particular mode of compounding the symbols ; this will be 

 the case even if the functional operations involved in the symbols 

 6, (f), &c. contain parameters such as the quaternion imaginaries 

 i,j, k ; but not if these functional operations contain parameters 

 such as the imaginaries which enter into the theory of octaves, 

 &c., and for which, e. y. a..^'^ is something different from u^.j, 

 a supposition which is altogether excluded from the present 

 paper. The order of the factors of a product 6(l>x • • must of 

 course be attended to, since even in the case of a product of 

 two factors the order is matei'ial ; it is very convenient to speak 

 of the symbols 6, (f) . . as the first or furthest, second, third, &c., 

 and last or nearest factor. What precedes may be almost entirely 

 summed up in the remark, that the distiibutive law has no ap- 

 plication to the symbols 6(f> . . ; and that these symbols are not 

 in general convertible, but are associative. It is easy to see that 

 ^=1, and that the index law 6'". ^" = ^'"+", holds for all posi- 

 tive or negative integer values, not excluding 0. It should 



* Communicated by the Author. 



