24 Mr. J. H. Cotterill on Elliptic Bibs. 



Substituting the value of M just now given, we find 



dV 



d¥ r 



= F a 3 cos 2 \ 2 sin 2 tfA'dtp + B a 3 cos /3 f \in<j)'{l - cos</>') AV/J 

 Jo Jo 



7T 



+ M « 2 cos/SJ 2 s in </>' A'#'. 

 Let 



7T 77 TT 



f sinVA'^^P; r 2 sin^cos(/)'A'#'=Q; f 2 sin</>'AW<ft' = K, 



Jo Jo Jo 



Then 



P = V 2 sin 2 <j>'A'd<l>' = ( 2 sin 2 iff V 1 - sin 2 £ cos 2 <£' ^ 



Jo Jo 



= I 2 cos 2 <£' ^l-sm 2 /3sin 2 <ftW(// 

 Jo 



J' 2 ~ cos 2 <£'- s i n 2 ft sin 2 <ft' cos 2 <ft ' 

 „ " TT-sin^sii?^' * 



_ T J 1 - (1 + sin 2 g) sin 2 eft' + sin 2 ft sin 4 if)' 

 J yl-sm 2 ftsm 2 <ft' *' 



Now by a known formula of reduction (Hymer's ' Integral Cal- 

 culus/ p. 191), 



f i T sin 4 ftgff _ 2 1 + sin 2 ft f if sin 2 0'ffi 



Jo j ^I^^Biii^smV ~ 3 " sin 2 tf J V 1 - sin 2 ft sin 2 d>' 



__J_f 2 ~_ 

 3sm 2 ftJ VI 



sin 2 ft sin 2 <ft 



^X/' 



Jo \/l-sin 2 ftsm 2 0' 



i +i^p ira^-- sin * f sin2 ^ , 



dn 2 ftj " VI- sin 2 ft sin* <£' *" 



3 sin 5 



If then the complete elliptic functions of the first and second 



