of Algebraic Equations of the Fifth Degree. 573 



'J 5. Again, if we consider that Py//|3,\, Py/ are rational 

 functions of P/c/3«)» I/> using R as the characteristic, we shall 



have 



^f'i^.f) ~ ^ ^/(^.O ~ ^ ^'^ ^/» 



and Py/^^,) = irP/'=^rRPy. 



Hence the equation for P will be such that 

 RirPy. = hRP/; 



and if we further consider that there must subsist an equation 

 analogous to (aa.) when each of the remaining roots is com- 

 bined with Py, we shall find ourselves conducted to another 

 class of equations solved by Abel in the memoir just alluded 

 to. 



46. Lastly, H/z/ss-) being (41.) a rational function of P///S8\ 

 we obtain (42.) 



X' indicating a rational function. 



The equation for the celebrated function H will therefore 

 belong to the same class as that for W, and must consequently 

 admit of a similar solution. Meyer Hirsch, in endeavouring 

 to solve this equation, alighted upon the equation of the 15th 

 degree analogous to V*^ + Cj V^'^ + . . =0. 



At some future time I hope to return to this subject, and to 

 discuss the resolution of the trinomial equation 



x^ + A^x + Agss 0; 

 to which very simple form the general equation of the fifth 



absolute or unconditional theorem of Taylor, we ought to have, as the dis- 

 cussion in the text first led me to perceive, 



where i may denote any term of the series 0, 1, 2, 3, . . . 



And thus, to take the first example which suggests itself, we see that 

 there cannot subsist 



((ai,)(^.!'-) = (i{'+T + v+-})W 



both when b'=a, and when S =b. For the series which constitutes a factor 

 of the equation of condition will be essentially divergent in one of these 

 two cases: so that this equation will no longer be satisfied in virtue of the 



other factor A(q). 



