244 Sir William R. Hamilton on the Argument of Abel, 



F/'"'" = (1, 2, 3, 4, 5);'" = a + 6 (^1 - ^.) • . • (^4 - ^5)' 



vr"" = (1,2,3,4,5)1" = a-b(x^~ x^) ... (x,- X,). 



These last equations show that the cube or fifth power (according as a," is 3 or 5) 

 of the product of (1, 2, 3, 4, 5),. and (1, 2, 3, 4, 5)^ is symmetric, and consequently, 



by what was lately proved, that this product itself is symmetric ; so that we may 



write 



f/' . f;" = (1, 2, 3, 4, 5). . (1, 2, 3, 4, 5), = c, 



and therefore 



V(l, 2, 3, 4, 5). . v(l, 2, 3, 4, 5)^ = c, 



V being here the characteristic of any arbitrary change of arrangement of the 

 five roots, which change, however, is to operate similarly on the two functions to 

 which the symbol is prefixed. (For example, if we suppose 



(1, 2, 3, 4, 5). = (1, 2, 3, 5, 4), (1, 2, 3, 4, 5), = (1, 2, 4, 3, 5), 



and if we employ V to indicate that change which consists in altering the first 

 to the second, the second to the third, the third to the fourth, the fourth to the 

 fifth, and the fifth to the first of the five roots in any one arrangement, we shall 

 have, in the present notation, 



V(l, 2, 3, 4, 5). = (2, 3, 5, 4, 1), v(l, 2, 3, 4, 5)^ = (2, 4, 3, 5, 1) ; 



and similarly in other cases.) Supposing then that y denotes the change of 

 arrangement of the five roots which is made in passing from that value of the 

 function f/' which is = (1, 2, 3, 4, 5\ to that other value of the same function 



which is = Pa"(l> 2, 3, 4, 5)^ we see that the same change performed on 



(1,2,3, 4, 5)^ must multiply this latter value not hy p „ but by p~\,; which 



factor is, however, of the form p^ „, so that we may denote the 2 a," values of 



F," as follows: 



(1, 2, 3, 4, 5). ; V d, 2, 3, 4, 5)^ ; . . . v"'"~' (1, 2, 3, 4, 5). ; 



(1, 2, 3, 4, 5),; V (1, 2, 3, 4, 5), ; . . . v°'"~' (1, 2, 3, 4, 5), . 



