Algebraic Equations of the Fifth Degree, 125 



We also have 



(/(Qi^Q^) =a-X^) -5S(^i V%) 

 and 



^(Qi^Qs) = «^' W H- 5S (x^^x^x.^x^ - 5 S 1 (a^i^i^-ga^g) + :^ (^) . e (a;) . 

 The last equation is a consequence of that of (73). There are, 

 of course, two more equations resembling the latter three. 



I have elsewhere shown (see pp. 226 and 486 of vol. lii. of the 

 Mechanics' Magazine) that when cr[x) is a symmetric product, 

 all such functions as cr(a7'^), <r (^), &c. are symmetric. The rule 

 which I gave in (6) of p. 299 of vol. i. of the ' Mathematician ^ 

 may, it would seem, be applied to the derivation of (t[x)j cr^ix), 

 &c. from a{x). 



For quintics wanting their second term ^(Qi^Qg) is simpler in 

 form, and equal to 



a (x) + ^X(x^Xc^2p'^^ — 5 S 1 {xi^XoX^) . 



84. Glimpses of a relation between different epimetrics have 

 already presented themselves. With that of (72) may be classed 

 the following : — 



X^X^XqX^X^ L.t^3^4 X^Q_J 



each side of which is a six-valued function. And there is also 

 an inexhaustible mine of relations corresponding to that of (73). 



85. By S^{x{^Xci''x/) I denote the epimetric 



Si { <!^r {Xc^X^P + ^5"^/ + ^3^^/ + x^^'x^p) } . 



86. Let 



then, from equations noticed in (57) and (58), we have 



^2 4 3 1 

 in other words. 



We also find 



B,=a,(^^) =^W, S6=S,(^*) =t'(r') 



and 



fg «4 «2 



CT "1 I *^3 , "4 I '^a . 



^^-r + r + r + r^ 



to I A La fci 



