1878.] relating to a quintic equation. 159 



the quintic equation in x is 



= (cc - cf 



-\- {x— cf .—5 (pr + qs) 



+ {x — cy. — 5 {p's + q^p + r^q + sV) 



+ (x — c) . — 5 {p^q + q\ + r^s + s^p) + 5 ( j) V^ + 5- V) — 5p^rs 



+ (a; - c)\ - (jj' + q^ + r' + /) 



+ 5 (pVs + (fsp + ^^^jjg- + s^qr) 



— 5 (j»^gV + q^r^s + r^s^j) + s^^^g*), 



and if we substitute herein for p, q, 'r, s their values, then, altering 

 the order of the terms, the final result is found to be 



X)={x-cY 



+ (a; - c)^ - 5 {AA^ + A^A^ aa^a^a^ 



+ {x-cf .-^ {A^Afl^a^+A^A^a^a + A^A^aa^ + A^Aa^a^ aa^a^a^ 



+ (ic — c) . - 5 {A^A^a^a^a^ + A^A^a^a^a+ A^^A^a^a\ 



+ A^A^aa^a^ aa^a./i^ 

 + 5 {A' A,' + A,' A,' - A^A,A,AJ (aa^a,a,y . 



+ {x — cy .— {A^a^a^a^-\-A ^aji^a^ + A ^a^a^a^+A ^aa^a^) aa^a^a^ 



+ 5 {A^A^A^a^a^ + A^A^A^a^a + A^A^Aaa^ 



-\-A^AA^a^a^[aa^a^a^'' 



- 5 {A^A^^A^a^a^ + A^^A^'A^a^a^ + A^A^Aa^a 



-f A^A^A^aa^) {aa^a^aj, 



viz. considering herein A, A^, A^, A^ as standing for their values 

 K-\- K'a-\- K"a^ + K"'aa^, &c. respectively, each coefficient is a 

 function of «, a^, a^, a^ unaltered by the cyclical change of these 

 values, and therefore a rational function of 



m, n, e, h, K, K', K", K"'. 



