respecting Equations of the Fifth Degree. 239 



two-valued form (II. III.). And if we suppose (1, 2, 3, 4) ~ (2, 1, 3, 4), in 

 order to reduce the six functions (1, 2, 3, 4) . , . (2, 1, 3, 4) to three distinct 

 groups, the functions (1, 2, 3, 4) and (2, 3, 1, 4) being supposed unequal; and 

 then require that of the six following values, 

 (1, 2, 3, 4), (2, 3, 1, 4), (3, 1, 2, 4), (1, 2, 4, 3), (2, 4, 1, 3), (4, 1, 2, 3), 



there shall be fewer than five unequal ; we must either suppose (2, 3, 1, 4) zz 

 (4, 1, 2, 3), in which case we are conducted to the three-valued form (I. II.) ; or 

 else must suppose (2, 3, 1, 4) = (2, 4, 1, 3), which conducts again to the four- 

 valued function (I. I.), by giving (I, 2, 3, 4) = (p{x^. 



Now of the four forms (a), (b), (c), (d), the form (a) is five-valued, and 

 therefore admissible in the present inquiry ; but the form (b) is, in general, ten- 

 valued ; the form (c) has, in generalj fifteen values ; and the form (d) has 

 twenty. If, then, we are to reduce the functions (b) (c) (d) within that limit of 

 number of values to which we are at present confining ourselves, we must restrict 

 them by some new conditions, of which the following are sufficient types : 



— G) i«I'4j *^\ **^2 • 1 *~" 3 • 1 6 • 2 '~~ 3 • 2 5 * 3 5/ ' 



(c)' . . (x^, X, x^-\- ^3 x^) = ^ (x^, X, x^ -f- 0^2 a;,) ; 



(c)" . . (x^, x^ X.-, -f- x^ x^ = (0^4, », a?2 -f x^ x^) ; 



(c)'" . . {x„ X, x^ -\- X, x^) = (x„ ir, X3 -{■ x^ X,) ; 



(d)' . . (x„ X,) = <p (x„ X3) ; 



(d)". .0(0;^, x^) = (p(x„ X3); 



{d.y"..<f>{x„ X,) = 0(0;^ JTa)- 

 (To suppose 0(^5, 0:4) = 0(^4, x^), would indeed reduce the number of values 

 of the function (d) from twenty to ten, but a new reduction would be required, 



VOL. XVIII. 2k, 



