respecting Equations of the Fifth Degree. 225 



which conduct respectively to functions with four, six, three, one, six, and two 

 values ; nor can any form of condition, essentially distinct from all the ten last 

 mentioned, be obtained by supposing any three or more equations to exist 

 between the twenty-four functions f,. 



A little attention will not fail to evince the justice of this enumeration of the 

 conditions under which a rational function of four arbitrary variables can have 

 fewer than twenty-four values : yet it may not be useless to remark, as connected 

 with this Inquiry, that, in virtue of the notation here employed, the supposition 



a,b b,a a,b ^ 



Vi = 1 involves the supposition Vi = 1 ; ^^ supposition Va = 1 involves the 



h,a c,d d,c a,b,c b,c,a c,a,b 



suppositions V2=l> V2=l> V2 = l; Vs = 1 involves Vs = 1» V3=l> 



a,c,b c,b,a b,a,c o,6,c b,c,d c,d,a d,a,b 



V3 = l, V3=l. V3 = l; V4 =1 involves V4 = 1> V4 = 1> V4= 1> 



a, c a,d,c d,c,b c,b,a b,a,d a,b 



V2 = l, V4 = 1, V4=l. V4 = l» V4=l; while the system Vi = 1> 



a,b a,b c,d a,b a,b,c 



V2 = I is equivalent to the system Vi = l, Vi = l; Vi = l> V3= 1» to 



a, b a, c a,b a,b,c a,b b,c,d a,b a,c,b 



Vi = i, Vi = i; Vi=i, V4=i> tovi=i, V3=i; Vi = i> V4=i» 



a, b a,c a,b a,b,c a,b b,d a,b,c 



to V, = 1, V2 = 1; V2 = i» V4 = i» to V2 = i' Vi = 1; V3 = i. 



bfCfd a, 6 a, ft, c a,hiC a, J, c ajbtC Cjd 



V3=l. to V2=l5 V3=l; V3 = l> V4=l, to V3=l» Vi = l; and 



a,b,c a,c,b a,b,c a,b,c 



V4 = l, V4=l) to V3=l> V4=l: analogous equivalences also holding 

 good for other systems of analogous conditions. 



Let us now consider more closely the effects of the ten different suppositions 



(I.), . . . (II. III.)- 



In the case (I.), the function f is symmetric relatively to some two roots 

 x^, x^, and may be put under the form of a rational function of the two others, 

 X , x^i or simply of their difference, 



(I.). ..F = 0(^^—0;^); 



it being understood that this function may involve the coefficients a^, a^, a^ , a^, 

 which are symmetric relatively to x^, x^, x„ x^: because it is in general possible 

 to determine rationally any two roots x , Xg, of an equation of any given degree, 



when their difference, x — ^^, is given. 



2 H 2 



