:.iv."\i'i-s.i\wn-;:r 



M- v.mv Jfy means of Symmetric Functions. ^' '^ ■ ^^ $S$ 



we shall perceive that Q may be completely detached IfQtft bpth 

 these equations if we determine P, R, S, T so that '"' '" 



5'"' ^ (5.(0P + 2R + 3S + 4T)=0, 



©1 . (OP + 2R 4- 3S + 4T) =0, 



@.(0P + 2R + 3S + 4T)2=0, 

 where the product of the numbers which mark the dimensions 

 relatively to P, R, S, T is only 1 . 1 . 2 or 2. 



In this way, without resolving any equation of a higher degree 

 than the second, we shall have 



A\=OQ + 0, 

 A'2=0Q2+0Q + 0. 



And Q, which as yet, therefore, is wholly undetermined, may 

 now satisfy the cubic equation 



v^ ,' . . g,(0P + lQ + 2R + 3S-f-4T)3^Q; 

 or rather 



g;^- 3@P.(0P + 2R + 3S+4T)Q2 + 



3@1 . (OP + 2R 4- 3S + 4T)2Q + . ,^y ^:^ 

 @.(0P + 2R + 3S+4T)3=0; 

 thus fulfilling the third and last condition 



A'3=0. 

 6. It naay easily be shown that P, Q, % 3, T, y will none of 



them assume the form 71— TTrfi where G and H are integral 

 (1 — 1)11 



functions of the (m— 2) arbitrary coefficients A3, A4, . . A^. 



Returniug to the equations in P, R, S, T, we see that the fir^t 



pf tjjem will be reducible to 



@.(0P + 3S + 4T) = 0, 



since ©2=0. We may also perceive that the second equation 



of the group will become 



@.(3R + 4S + 5T)=0. 



For, since in general ©ti'=©t@i' — ©(t + v), it is evident that 

 the coefficients of P, R, S, T, in 



@1.(0P + 2R + 8S + 4T) 

 may all of tljem be derived from the expression 



©1@u-©(1+v), 

 on taking v successively equal to 0, 2, 3, 4 j whence 



©1.(0Ph-2R + 3S4-4T) = -@.(3B' + 4S + 5T), 

 ©1 being equal to zero. 



