248 Mr. J. V. Jones. On the Calculation of the 



ff0 -p .J • J? 1.3.5... . (2m- 1) 1 / a; V M -d 



§ 2. Putting K m = ■ — I ) P m , . 



s to -2.4.6. ...2m 2m-f-l\A + J 



so that M© = ®(A4-a)c 8 2(-l)«K w 



it is now farther proved that 



2m(2m + l) , 8 f (2m-l)(2m -3) 1 



~ (2w + 2)(2m + 3) \ ^ m "~ 2m . 2m 6 JW " 1 f 



, 1 + c' 2 / * \« 



where a. = — 



A + a 



1 / x 



e 2 = 



1 + c' 2 \A-f ay 

 c' 2 = 1-c 2 . 



Hence we see that the series is a recurring series. The relation 

 above given between K m+1 , K,», and K m _ x materially facilitates the 

 calculation of successive terms of the series. 



§ 3. If we put 



2(-l)"K» = W 

 2(-l) m mK tl! = T 



2m— 1 



dM dk. da dx 



aud m = ?a +7 v +5 it 



then _ 1-g T+2Y 



q ~ ' 2 + deW 



1-s T + 2Y 



2 deW. 

 I = 2T/W. 



§ 4. The following is a more general expression of M© in terms of 

 elliptic integrals applicable for all values of x : — 



[F— E c' 2 ~l 



where 



2x/Ac 



F = 



E = 



(A + a) 2 + ar" 

 J Vi^-fc 2 si 



ii 2 <9 



sin 2 . d6> 



