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



(iv) If the law holds good in the case of any series, it will 

 hold good also for its derivative, for 



k '& r ( log i - s ) *" = r ■ 2n ( l0 s i - B ) k " - rk ™ 



§ 6. If we start with the series for K' and E' in both of 

 which the law holds good, we may deduce the series for J' and G ' 

 by means of the formulae, 



J' = K'-E', 



Q' = E' -FK', 



respectively. The series for J' and G', derived from those for K' 

 and E by these equations, afford examples of the application of 

 theorem (ii). The series for J'—G', derived from those for J' 

 and G', affords an example of (ii), and the series for E'+ G' and 

 E'—J', derived from those for E', J', G', afford examples of (i). 



The series for K' and E may be derived from those for G' 

 and J' by means of the equations 



It= ~k'dk' h ^~ k Hk'> 



the series so derived afford examples of (iv). 



The series for J' and G' may be derived from those for E' and 

 K' by means of the equations 



the series for J' and G' so derived afford examples of (iii) and (iv) 

 combined. 



The theorems (i), (ii), (iii), (iv) serve to explain the occurrence 

 of the law in certain of the series, as derived from others ; but 

 it is none the less remarkable on this account that the law 

 should hold good in all the seven series, as certain restricted 

 conditions with respect to the coefficients have to be satisfied 

 in order that the theorems may admit of application. 



The Notation, § 7. 



§ 7. The meanings assigned to J and J' by Weierstrass* were 

 J=K-E, J' = E'. Thus Weierstrass used J' in place of E'. 

 As however it is convenient always to denote by accented letters 

 the same functions of /*;' that the unaccented letters are of k, 

 I have retained E' and used J' to denote K'— E'. 



* Crelle's Journal, Vol. lii. p. 361. 



