556 Mr. G. B. Jerrard's Rejections on the Resolution 

 there will arise 



as will be evident on reflecting that 



or that every substitution of the nXh degree, n being any num- 

 ber, may be represented by a substitution of the (« — l)th 

 degree followed by an interchange. 



19. I proceed to consider some properties of the function 



which in accordance with the meaning usually attached to the 

 symbol ( . . ), I shall express by 



x(«/3)(r.O"; 



^ab^ thus denoting the same thing as | ?• ). 



20. Beginning with the function X O^^^t it will at once be 

 seen that 



X(a/3) = X(^a); 



C^l^^ and (^'"■^ being equally expressive of an interchange 

 of the elements a and jo. 



21. Passing to the function X (*,^} C^..^}' ^^ ^^® ^^^^^ ^^ 

 a, /3, y, 8 be unequal, we must have 



X(«/3)(y8) = X('y8)(a^) («.) 



For the interchanges being, according to this hypothesis, in- 

 dependent of each other, it must be indifferent in what order 

 they are taken. 



But if the function in question were of the form X(^«^)(^^y), 

 in which the element /3 is common to the interchanges, we 

 should, exclusively of particular cases, alter the value of the 

 function by inverting the order of the interchanges. 



Thus if X = ^ (a, /3, y), 



we shall have X(a /3) (/3y) = ^ (y, a, /3), . 



and X(/3 y) («>) = ^(a, y, 0} («f ), 



/3' being arbitrary. 



Now, if ^ (y,*a, fi)='^ (a, y, /3) («f ), we must take /3' = y. 

 There will consequently result 



X(«i)(^.r) = x(^.y)(^.y); . . . (/3.) 



