respecting Equations of the Fijlh Degree, 179 



transformation, the radicals a, , . . . . o . -, may all be removed from the deno- 



(m) 



minator of the rational function f ; and that their exponents in the transformed 

 numerator may all be depressed below the exponents which define those radicals : 

 by which means, the development above announced for the general irrational 



(m) _ . , (m — 1) 



function h may be obtained ; wherein the coefficient 6 , , r™-) admits of 



being analogously developed. 



For example, the function of the second order, 





^"=-i + <+a. 



which was above assigned as an expression for a root of the general cubic equa- 

 tion, may be developed thus : 



5" = S . (v . af' ) = b; + h; a," + 6,' a/' ^ 

 /3,"<3 V A" / 



in which 



»/ ^l_ J I 1 7/ ^2 £2 ^2 



Oo — — 3 . 0, — A. O2 — Oj"3 -f^' — c^^a,' 



And this last coefficient b^, which is itself a function of the first order, may be 

 developed thus : 



*^' = e^ = B'= E . (b . a/') = B„+ B.a/ ; 



in which 



__ C2C1 __ CjjCi C2C1 Ci 



Bo - e.-.-a,'^ - ^2_j-^ — -^— ^2, 

 -1 



Again, the function of the third order, 



aC'ai"' 



