340 Sir William Rowan Hamilton on Equations of the Fifth Degree. 



and then a corresponding value of X by the condition kX = i, we shall have ± )3 

 by extraction of another square root, since j3' = e -|- a- -f- X ; and may afterwards, 

 by the extraction of a third square root, either find ± y from the expression 

 y^ =z e-{- 6k -\- 6-\, and deduce 8 from the product ^yh =. t), or else find 

 — (7 "f" ^) from the expression 



{y + if=2e-K-X + ^; (18) 



and may then treat oc", x'", .x'*', x^, as the four values of « -}- /3 + 7 + 8, while 

 x" ^ — 4«. Hence any function whatever of the five roots of the general equa- 

 tion (1 ) of the fifth degree may be considered as a function of the five quantities 

 A, a, e, t;, t ; and if, in the expression of that function, the values (16) be substi- 

 tuted for a, 6, t], I, so as to introduce in their stead the quantities x', p, g, r. It Is 

 permitted to make any simplifications of the result which can be obtained from 

 the relation (2), by changing a/* -\- pi^'^ + (l^^-\- ''•^'j wherever it occui-s, to the 

 known quantity — s. 



14. Consider then the twentyfour-valued function, referred to In a former 

 article, and suggested (as Lagrange has shown) by the analogy of equations of 

 lower degrees ; namely, t% in which 



t zz x^-\- wx^ + MV3 -|- to^x^ + f^^^.v (19) 



and 



«* + «.' -f ft)^ -f « -f 1 = ; (20) 



a) here (and not a) denoting an imaginary fifth root of unity, so that 



«*=1. (21) 



Observing, that by (4) and (20), x^, &c. may be changed in (19) to x', &c. ; and 

 distinguishing among themselves the 1 20 values of the function t by employing 

 the notation 



4»erf, = «,V' -f w^x^'^ -f «.V=> + a.V"' -j- u,'x^'\ (22) 



which gives, for example, 



^■2345 = ^ + « V + «V" + wV + wx ''; (23) 



we shall have, on substituting for x' Its value — 4a, and for x", x'", x'^, x" 

 their values (7), the system of the twenty-four expressions following 



'& 



