respecting Equations of the Fifth Degree. 193 



which give 



116 /6 12 /2 /2 /3 Hi 113 



^i, 1, 2, 3 = *i. 1. 2. 3 /i /2 /a /4 03 a^; 



T, — C, O, , Tj — C2 a^ , T3 — C3 03,14 — C4 O4 , 

 if we put, for abridgment, 



« //» /,-10 , ,10 ,'"'" /.'' .'* '^ 



'^1 ^ ^2, 1, 3, 4 ^1, 1, 2, 3 5 *-'l —^ ^2, 1, 3, 4 "l,\,% 3 J I J 3 J i't 



It «— 10 «20 / /— 10 y20 /3 ,2 ,4 



'^2 ■— • *2, 1,3, 4 *1,1,2,3 5 ^2 —- "2,1,3,4 "l,l,2,3 ^2 J 3 J i ^ 



„ „18 ,,—24 , ,18 ,—24 /4 ,—2 , 



'^3-^^2,1,3,4 *l. 1,2,3 5 *^3 -~ "2,1,3,4 "l,I,2,3 J\ J 2 J 3 '1 



„ ,,—12 „18 , ,-12 ,18 ,—2 ,2 , 



"^4 -— ^2,1,3,4 ^1,1,2,3 5 '-'4 -^ "2,1,3,4 "1,1,2,3 J I Ji J i' 



And, with a little attention, it becomes clear that the same sort of process may 

 be applied to the terms t of the development of any irreducible function 



(m) 



b ; so that we have, in general, a system of relations, such as the following : 



(m) (m — 1) (m) ' (m) (m — 1) (m) 



' ' ' ' (m) (m) (m) 



, n B » 



. (m) . 



in which t. is the product of certain powers (with exponents positive, or nega- 



(m) _ (m — 1) 



tive, or null) of the various terms t , ^ ; and the coefficient c is difFe- 



(8,W,... 



rent from zero, but is of an order lower than m. For if any radical of the order 

 m were supposed to be so inextricably connected, in every term, with one or more 

 of the remaining radicals of the same highest order, that it could not be dis- 

 entangled from them by a process of the foregoing kind ; and that thus the 



W 

 foregoing analysis of the function h should be unable to conduct to separate 



expressions for those radicals ; it would then, reciprocally, have been unnecessary 



to calculate them separately, in effecting the synthesis of that function ; which 



function, consequently, would not be irreducible. If, for example, the exponents 



a/*"^ and ttj^, which enter into the equations of definition of the radicals. a/"^ 



and O2 , should both be = 3, so that those radicals should both be cube-roots of 



2d2 



