of the Fifth Degree. 363 



and conceive the four suffixes I, m, n and r to be permuted in 

 every possible way. Each of the twenty-four equations, 



.&jW=o, ^2^=0, ... v'-=o. 



thus obtained aflFords a different value of P, for no two permuta- 

 tions of four quantities can belong to the same quinary sequence. 



30. The expressions evolved by ]3ermuting a, b, c and d in 



are the same as those deduced by permuting y^, y^, y^ and y^ 

 respectively. If, therefore, recurring to arts. 15 et seq., we make 



%'l + ^ym-^^yn + ^yr = ^^^y ^> ^' ^) 



and ^= ^j X <^2 X ^3 ^ ^4' 



we shall be conducted to an equation of the sixth degree which 



may be represented by 



^ + Di^HD2^+. -.+1)6=0, 

 and in which D are symmetric functions of a, b, c and d, and, 

 consequently, of the four roots y^ y,„, y^ and y^. 



31. Now, assuming a = l and so identifying (}) with d, it is 

 easy to express D as symmetric functions of all the roots. For, 

 since no two permutations of four quantities can belong to the 

 same cycle of five, each and all of the expressions 



^W, d('\ ^(»", d'"', ^^"^ 

 indifferently, can be made to furnish the twenty-four values of P. 

 Let us adopt them in succession. We find, D denoting any one 

 of the coefficients of the sextic in 6, 



D = '»/r(«, b, c, d) = f{l, m, n,r) 



= ^(m, n,r,li)=. . =^fr{k, I, m, n), 

 and, therefore, 



5J) = 1, ."^{k, I, m, n). 



Consequently, since i/r is symmetric with respect to four roots, 

 D is a symmetric function of all the five roots y^, j'^, . • , in other 

 words, of y^, y^, .., y,. 



Section V. 

 33. Making 



^i{efff)=yAib+^)yf+{o + bH^yj,. 



where ^ is any root of 



and bearing in mind that 



F<(/,*) = F<(./.)(?)C*)(-^^). 

 we may recapitulate our principal results as follows : — 



