respecting Equations of the Fifth Degree. - 213 



they may also, on account of the mode in which they are formed from one com- 

 mon type F (x^, 1' ), be said to be syntypical functions. For example, the two 

 values, 



ax^-{-hx, = {\, 2) = f(^,, x^) = F,(^,.^,) = (l, 2),, 

 and 



ax. + bx^ = (2, 1) = F (x^, or,) = f, (x^, xJ = (1, 2)., 



of the function ax^ -\- bx^ , may be considered as being two different but syn- 

 typical functions of the two variables x^ and a:,. And again, in the same sense, 



the functions — and — are syntypical. 



X-i Xi J J L 



Now although, in general, two such syntypical functions, f^ and f., are un- 

 connected by any relation among themselves, on account of the independence of 

 the two arbitrary quantities or, and x^ ; yet, for some particular forms of the 

 original or typical function f,, they may become connected by some such relation, 

 without any restriction being thereby imposed on those two arbitrary quantities. 

 But all such relations may easily be investigated, with the help of the two gene- 

 ral forms obtained in the thirteenth article, namely, 



F^ = a-\-b(x^—x^),F^ = a — b(x-xJ, 



in which a and b are symmetric. For example, we see from these forms that the 

 two syntypical functions f^ and f. become equal, when they reduce themselves 

 to the symmetric term or function a, but not in any other case ; and that their 

 squares are equal without their being equal themselves, if they are of the forms 

 ± b (x— xj, but not otherwise. We see, too, that we cannot suppose F=P3 f,, 

 without making a and b both vanish ; and therefore that two syntypical functions 

 of two arbitrary quantities cannot have equal cubes, if they be themselves 

 unequal. 



[16.] After these preliminary remarks respecting functions of two variables, 

 let us now pass to functions of three; and accordingly let f(x^, Xg, x ), or 

 more concisely (a, j3, 7), denote any arbitrary rational function of any three arbi- 

 trary and independent quantities x^, x^, x^, arranged in any arbitrary order. It 

 is clear that this function f has in general six different values, namely, 



