SIR G. GEEENHILL ON ELECTROMAGNETIC INTEGRALS. 41 



(10) T = 4.^-t.ti-t.t-t,, t 2 >t>t.,, 



a a -A 2 



MQ a r -t 



, 9} 2 



MQ 2 PQ " r-t 



(13) -U = 4.*! r.T-,.T- 8> t 1 >T>t,>t a , 



2 -A 2 6 d0 _ p 2y x (-U) eft 



o MQ 2 PQ It. r-t v/T' 



in a standard Weierstrassian form of the III. E.I. 



The expression of this TIL E.I. when complete, by means of the E.I., I. and II., 

 complete and incomplete, was given by LEGENDRE, ' Fonctions Elliptiques,' chap. 23, 

 and (14) falls under his class (m'). 



4. But we shall avoid the Legendrian form, and start by making use of the 

 lemma 



proved immediately by effecting the differentiation. 



Integrating, in either order, with respect to the differential elements 



dt T dr 



and 



v/T 

 and between the limits t 2 > t > t z , and t lt T of T, 



(2) P *r (** ^T^t^T 



-< 



~ Iv/TV r-t I 



- 



