158 Prof. Cayley, On a theorem of Abel's [March 11, 



= 2a™/S'"'*"' (7'^"7''«+' + 722+iy 2p+i j , 



The second, third, and fourth expressions contain the factors 

 ^ (7^ "" y'^> 77' (7^ ~ 7'^)' l^yy respectively ; and the first expression 

 as it stands, and the other three divested of these factors respec- 

 tively ai'e rational functions of a, /8^, 7^,7'^, that is, they are rational 

 functions of m, n, e, h. But the omitted factors yS (7^ — 7'^), 

 77' (7' - 7"). /377', = 2nh (1 + e"), 2/i'e (1 + e'), oihe (1 + e') are ra- 

 tional functions of n, h, e; hence each of the original four expres- 

 sions is a rational function of m, n, h, e; and the entire function 



(f) {a, a^, a^, a^) + (f> (a^, a^, %, a) +^ {a.^, a^, a, aj + ^ {a^,a, a^, a^) 



is a rational function of m, n, h, e. 



Replacing a, /S, 7, 7' by their values, the roots of the quartic 

 equation are 



m + n^{l + e') + ^[h {\ + e" + ^J{l + e^))], 



m - w V(l + e') + ^/[h (l+e^- V(l + e'))], 



m + n V(l + e') - ^[h (1 + e' + V(l + e'))], 



• m-n V(l + e') - ^J'[h (1 + e' - V(l + O)]- 



And I stop to remark that taking m, n, e,li = — \, + \, 2, — | re- 

 spectively, the roots are 



-i + W5 + V[-H5+v/5)], 



-i-iN/5 + V[-i(5-v/5)], 



-i + i^/5-V[-M5^-^/5)], 



-i-ij5-V[-i(5-V5)], 



viz. these are the imaginary fifth roots of unity, or roots r, r^, r*, r' 

 of the quartic equation cc* + a?^ + ^^ + a; + 1 = ; which equation, as 

 is well known, has the group rr^r^r^, ^2^*^.3^^ r^r^rr^, r^rr'^r*. 



Reverting to Abel's expression for x, and writing this for a 

 moment in the form 



oc=c+p + s + r + q, 



