of Algebraic Equations of the Fifth Degree. 561 



f^ denoting whaty^ becomes when a^ is changed into a^^', we 

 shall readily perceive that the six groups of equations on 

 which, as we have seen, all the different expressions for P 

 depend will be reducible to the three following groups, 



/, =0,1 ^1 = 0,1 i\ = 0,\ /'i = 0,\ 



iu^Oj /„(25X = 0,J i'„=0,/ //„ (25)^ = 0,/ ^''^ 



^1=0,1 ^1=0,1 k\=^o,\ ^^'i=o,n ,u^ 



K=Oj ^.(35)„ = 0,/ /<:'.= 0, //.'„ (3 5)^ = 0, J (''•^ 

 l^ = 0,1 I, = 0,1 /', = 0, 1 l\ = 0,1 



^.=0,J /„(*?)« = 0J ^'.=0,/ /'^ (45)^ = 0,/ ^'-^ 



C^0»» C^^)«' C*^)w "^^y denote either the three inter- 

 changes (25), (35), (45), or (34), (24), (23), the com- 

 plementary interchanges relatively to fj, k^, ly For greater 

 uniformity the same index o has been retained throughout the 

 groups; the expression for P being in every case unaffected 

 in value by writing 2, 3, 4, 5 successively instead of y. 



28. But for our purpose it will not be necessary from each 

 pair of these equations actually to find an expression for P in 

 terms of .rj, a^g, . . cc^. In effect, if we denote by P/zab) Ccd) 

 that value of P which is derived from the pair of equations 

 /« (^.!^) (^.^0 • • = 0, /„ (ab) (cd) . . = ; the twelve values 

 of which P is susceptible will be represented by 



Pi. Pi(25)„. Pj'. Pi' (25)^^ 

 P^. P^(35),,. P^> ^k'(Zhy 



P/» P/(45),. P/'5 Pi' (45),,: 



so that without proceeding any further we may perceive that 

 the roots of the equation P^'^ + Bj P^^ -f . . = must be such 

 as to admit of being distributed into three groups which are 

 related to each other in a very remarkable manner; the second 

 pair of roots in each group being derivable from the first pair 

 by merely introducing a^2 instead of a^ . Indeed the groups 

 themselves may all of them be derived from 



P/» P/(/5.o,.' P/'' P/'(^.0„' 



which will represent four roots of the equation for P. 



Section V. 

 29. At first view it might be imagined that, if 

 Vp = P^ + P/(/3.),, + P/' + P/'(/J.)„» 

 Phil. Mag. S. 3. No. 176. Suppl. Vol. 26. 2 P 



