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ROYAL SOCIETY OF CANADA 



As the ordinary tables have not been calciilated for a weight 

 higher than fifteen, so these tables have not been calculated for 

 weight higher than fifteen. 



4. Since -^' ^ = ^ «, «, 



^ c = ^ a a a 



etc. 



etc. 



the coefficients of these functions in the present tables must coincide 

 with those of the corresponding functions in the ordinary tables. As 

 a further check to the work the laws given' by the author have been 

 used for the ground they cover. The most complete lest, ^ however, 

 has been by means ©f the operator due to Briochi, 



+ Px 



d 



+ + />n-l 



d 



d 



dPi ' ^' dpz '" " dpn d s^ 



Let the left hand member of this operator be denoted by P. 



5. If we have a symmetric function of the roots ^i, X-j, . 

 which expressed in terms of the coefficients is / (/>,, Pi . . . 

 and in terms of the roots is F^ {s\ s^ . . . . s]), where s\ 



1 a <' » 



can easily find the derivative with respect to s^. For if 



F'{s],s\ .... si) = F{s„s, .... s^), 



^'/fc' 



Pn) 



we 



where s^ = 2 a', then, since 



.<?j = 2 k^ = 2 a^ «J «3 



«k = Pk 



k! 





 2 



.«, 3 



to differentiate F^ (s\, s*, s' ... . «J) with respect to s\ is equivalent 

 to differentiating with respect to 



k! 



s, 1 

 sj s^ 2 



Sk 



1 Metzler — Some Notes on Symmetric Function?, Proc. Lond. Math. Soc. 

 Vol.XXVlII.No. 609. 



2 In addition to these tests many of the résulte have been verified by Mr. 

 0, S. Stetson and Mr. D. Pratt of the mathematical department of Syracuse 

 Univert^ity. 



