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



in which a and b are symmetric. It is evident that any farther diminution of the 

 number of values of F, conducts, in this case also, to the one-valued or symmetric 

 function a. 



Combining the foregoing results, we see that if an unsymmetric rational 

 function of three arbitrary quantities have fewer than six values, it must be 

 reducible either to the two-valued form 



a + b (x-x,) (^ -iCj) (:v^-x,), 



or to the three-valued form 



a + bx + cx^. 



[18.] It is possible, however, that some analogous but different reduction 

 may cause either — I. the square, or II. the cube of a function f of three variables, 

 to have a smaller number of values than the function f itself. But, for this pur- 

 pose, It is necessary that we should now have a relation of one or other of the 

 two forms following, namely, either 



I (a^, ^2. 72) = — («i> A> 7i) 



or 



II. . . . (a^, p,, 7,) = P3 (a„ ^„ -y,), 



(Pj denoting, as above, an imaginary cube root of unity,) instead of the old func- 

 tional relation (a^, /S^, 7^) =z (a„ /3,, 7 J. And as we found ourselves permitted, 

 before, to change that old relation to one or other of these two, 



1st, (|3, a, 7) . = (a, ft 7). ; 2nd, (/3, 7, a). = (a, /3, 7). ; 



so are we now allowed to change the two new relations to the four following : 



I. l,..(ft 0,7).= -(a,|8,7).; I. 2, . . (/3, 7, a). = - (a, /3, 7).; 

 II. 1, . . (A a, 7). = ft (a, ft 7). ; II. 2, . . (ft 7, a), = p, (a, ft 7). ; 



the relation (I.) admitting of being changed to one or other of the two marked 

 (I. 1) and (I. 2); and the relation (II.) admitting, in like manner, of being 

 changed either to (II. 1) or to (II. 2). But the relations (I. 2) and (II. 1) 

 conduct only to evanescent functions, because (I. 2) gives 



