[metzler] some new SYMMETRIC FUNCriON TABLES 



27& 



But to differentiate with respect^to this is equivalent to differentiate 

 ing with respect to .Vj and dividing by 



s, 1 . . . 



1 



(^ - 1) ! 



2 



Sfc-l 



It follows therefore that : 



♦»»-! 



etc., etc. 



6. vSince the labor of finding the symmetric functions of the 

 type 2 k^ k^ . . . . kji b}^ first expressing them in terms of 

 symmetric functions of the a's, and then by the ordinary tables in 

 terms of the ])'s, is very great, some short cut or direct method of 

 obtaining them would be very desirable. Such a method I shall 

 illustrate in the case of finding 2 a^ a^ a^ . . . . a^. 



7. If n = 6 there are «C", = 15 products of the roots two at a 

 time, and having -5" Oj Oj . . . a, we can, by one of the laws referred 

 to in art. 4, write down all these terms in ^ «j a, . . . a^ contain- 

 ing no p with a subscript greater than six by taking the terms 



+ Pb {P\ Pi Pi - P\ Pi + Pi P2 Ps + 3 P3 ;>5 - "2 pI Pi - Pi P3 Pi + 



P'^ Pe + P\) + Pf, {v\ Pt + Pi Pi - '^Px P% Pi - Pi PÙ from 2aia^ . . a„ 

 changing each p into its complementary {p^ and p^, p^ and p^, p^ and 

 />3 are complementary) and multiplying, where necessary, any term 

 by a proper power of p^ to raise its weight to sixteen. Thus we get 



+ Pi iPl Pi P5 + ^ Pi PZ Pi - '^ P-i Pi - P2 PZ Pi + pI + P' P9 + P\ Pi 



+ Pi p\ Pi - '^pI P% Pi - p\ Pi Pe) + pI ijh Pi - Pi Pi)- 



Again if we observe 2 a, a,, 2 a„ a.^ a^ 

 it will be seen that they may be written 



a, «2 



Pi Pi 

 Pi 1 



Sec. III., 1908. 18. 



