188 Mr Glaisher, On the developments of [Nov. 24, 



§ 4. The coefficient of h 2n in each of these seven expressions is 

 of the form 



±r ( log & _i 7' 



where r is a fraction whose numerator and denominator are 

 products of factors, and R is a numerical quantity, the connexion 

 of which with r is best exhibited by assigning to n a particular 

 value. Taking, for example, n = 3, the values of r are 



1 2 .3 2 .5 2 



(i) K', 



2 2 .4 2 



r.3 2 .5 



W ^ > 2 2 . 4 2 . 6 ' 



1 2 .3 2 .5 



^mj J , 2 2 4 2 6 2 ' 



(iv) G', 

 (v) /'-£', 



r.3 2 



2 2 .4 2 .6' 



r.3 2 



2 2 .4 2 .6' 2 ' 



(vi) E'+G\ ^rrQ> 



, ... _ r , 1 2 .3 2 . 5. 7 

 (vn) A -J , 2 \4<\ 6 2 



and the corresponding values of R are 



V) 1 2 ~ 3 4 ~ 6 6> 



V s t 1 2 T 3 4 ~ 5 B> 



1 2 



(iii) 



(iv) f - | + | - I - i, 



(v) f-i + f-f-f, 



(vi) f - | + f - i - h 



(m\\\ 2_2t2_2il_2_L 



I — 2 4. 1 



T 



In all cases the quantities r and R are so connected that to every 



factor m^ in r there corresponds a term + — in R, the sign being 



positive or negative according as m** occurs in a numerator or 

 denominator. 



