respecting Equations of the Fijih Degree. 253 



veloped In this paper might have shown beforehand that he must necessarily fail) 

 in his endeavour to calculate the latter two, or the former ten functions, through 

 any finite number of extractions of square-roots, cube-roots, and fifth-roots of 

 expressions of the form a + \/—i b. 



[28.] But when we come to consider in what sense it is true that we are in 

 possession of methods for extracting, without tentation, such roots of such imagi- 

 nary expressions ; and therefore in what sense we are permitted to postulate the 

 extraction of such radicals, or the determination of both x and y, in an imaginary 

 equation of the form 



X 



4- V—iy = V a + V— 1 by 



as an instrument of calculation in algebra ; we find that this depends ultimately 

 on our being able to reduce all such extractions to the employment of tables of 

 single entry : or, in more theoretical language, to real functions of single real 

 variables. In fact, the equation last-mentioned gives 



{x-^-x/ZTiyfzza-^V-b, 



that is, it gives the system of the two following : 



which, again, give 

 and 



a_l afa— 1) (a — 2) a—Z •. i o i 



a^ y — rrrr 3 ^ y + &c. = 6 ; 



{x^ + f) =a^ + 6^ 



at— a(a— l)(a-2) /yV 



X 1.2.3 \xJ ^ _ b 



aia-l) fyy , -a' 



~ 1 . 2 



©*+... 



If then we put 



0. (P) = P"' 



