1884] K', E', J', G' in powers of the modulus. 189 



The law of the coefficients, §§ 5, 6. 



§ 5. With respect to this curious law the following theorems 

 may be noticed : 



(i) if the law holds good in the case of two series in which the 



4 

 factors multiplying k 2n log y are r and ra respectively, it will hold 



good in the case of the sum and difference of the two series, for 



vol 



and 



(* + l) + r B-,.<, + l) (* + _!-), 



The case in which the corresponding multipliers are r and r - 



is obviously identical with that just considered, but the inde- 

 pendent proof is worth noticing, viz. 



rR-rjR-l) = r^±(B+ 1 1 



13 J £ V 13-1 J3J> 



(ii) if the law holds good for two series in which the cor- 

 responding factors are r and r -3 , it will hold good for their differ- 



P 

 ence, if /3 = a + 1, for 



(iii) if the law holds good for two series in which the cor- 

 responding factors are r and r -~ it will hold good also 



for their difference, for 



It is obvious that in this theorem we may suppose the cor- 



a 2 



responding factors to be r and r : . 



(a-l)(a + l) 



