respecting Equations of the Fifth Degree. 215 



so long as the form of the function r, although It has been supposed to be 

 rational, remains otherwise entirely undetermined. But, for some particular 

 fonns of this original or typical function F(a:„, a;^, x ), relations may arise 

 between the six syntypical functions Fj, . . . Pg, without any restriction being 

 thereby imposed on the three arbitrary quantities a:,, x^, x^; for example, the 

 function f may be partially or wholly symmetric, and then the functions Fi, . . . f^ 

 will, some or all, be equal. And we are now to study the chief functional con- 

 ditions, under which relations of this kind can arise. More precisely, we are to 

 examine what are the conditions under which the number of the values of a 

 rational function f of three variables, or of the square or cube of that function, 

 can reduce itself below the number six, in consequence of two or more of the 

 six syntjrpical functions f,, . . . Fg, or of their squares or cubes, which are them- 

 selves syntypical, becoming equal to each other. And for this purpose we must 

 first inquire into the conditions requisite in order that any two syntypical func- 

 tions, or that any two values of F, may be equal. 



[17.] If any two such values be denoted by the symbols 



t(x^, Xp^, x^j, and f(x^^, x^^, x^), 



or, more concisely, by the following, 



(a , ^,, 7,) and (a, /3^, yj, 



it is clear that in passing from the one to the other, and therefore in passing from 

 some one arrangement to some other of the three indices a, ^, 7, (which must 

 themselves coincide, in some arrangement or other, with the numbers 1, 2, 3,) 

 we must have changed some index, such as a, to some other, such as ^, which 

 must also have been changed, itself, either to a or to 7 ; this latter index 7 

 remaining in the first case unaltered, but being changed to a in the second case. 

 And, in whatever order the indices Oj, /Sp 7i may have coincided with a, j3, 7, it 

 is obvious that the function 



must coincide with the syntypical function 



F^(ar„, x^ x^) or (a, ft 7)., 



VOL. XVIII. 2 G 



