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



a'* ; namely, by putting that which had been 6'* in the place of that which had 



a, h,c 



been a'*, and so on ; and finally V45 that the a'* is to be changed to the 5'*, the 

 6** to the c'*, the c** to the rf'*, and the rf" to the a'* : so that we have, in this 

 notation, 



The first sort of change may be called, altering in a simple binary cycle ; the 

 second, in a double binary cycle ; the third, in a ternary ; and the fourth, in a 

 quaternary cycle. And every possible equation, 



(a2» ^2» 72» y = («P A> 7i> y. 

 between any two of the twenty-four syn typical functions f^, may be denoted by 

 one or other of the four following symbolic forms, in each of which the two mem- 

 bers may be conceived to be prefixed to a function such as (a,, /3„ 7,, 8,) : 



a,J) a,b a,b,c Othc 



I...v, = i; II...V2=i; III... V3 = i; IV...V4 = i; 



or, without any loss of generality, by one of the four following, in each of which 

 the two members are conceived to be prefixed to a function such as (a, /3, 7, B) . : 



1,2 1,2 1,2,3 1,2,3 



I. .. v. = i; II. ..V2 = i; III. ..V3 = i; IV. ..V4=i; 



the I" and IP" suppositions conducting to twelve-valued functions, the III'" to 

 an eight-valued, and the IV"* to a six-valued function ; while every possible pair 

 of equations between any three of the same twenty-four syntypical functions, if 

 it be not included in a single equation of this last set, may be put under one or 

 other of the six following forms : 



1,2 1,3 1.2 3,4 



(I.I.).. V,= 1, V,= 1 

 (I.II.)..V, = 1, V2=l 

 (ILII.)..v. = l, V2 = l 



(I.I.)' .. V. = 1, V, = 1; 



1.2 2,3,4 



(I.III.).. v, = i, V3 = i; 



1,2 1,2,3 



(II.III.).. v, = i, V3 = i; 



