320 Prof. G. H. Darwin. On Jacobis Figure of [Nov. 25, 

 c=acos7, sina= /i /— — -, and 5=acos/3 



V Or — C 2 



(3) 



so that sm/3=sinasin7, and b = a\/ (1— sin 2 a sin 2 7.) 



l — c? sin 2 7 



Also let A 2 =u + d 

 whence B 2 =u + b 



sin 2 

 a 2 sin 2 ^ 



sin 2 



sin 2 # 



^-(l-sin 2 * sm 2 6>)> 



a 2 sin 2 7 ,, n 



sm~6> 



and 

 and 



2a 2 sin 2 



sin 3 



-cos e de, 



(4) 



r^=2a 2 sin 2 7 [ --?^^=2a 2 sin 2 7r C -?^^7. 

 Jo Jo sm 3 Jo sm3 7 



Lastly, let A = -/(l — sin 2 * sin 2 7) , 



and in accordance with the nsual notation of elliptic integrals let 



•H> E =J> 



Then we have the following transformations : — 



(5) 



-r 



Jo 



w r 



ada ~ J 



i 



c£t£ 



ABC a sin 7 



F 



(Ztt 



2 fsin 2 7 7 2 _ 



= 3 • 3 = 3.0 . 3 (F-E) 



cdc"~! ^B£ 3 



* Y sin 2 7 



>■ • (6) 



J.J3 3 C f a 3 sin L 7 j 

 c£tt 



2 rtan 2 7 

 — a 3 sin 3 7j A 



3 ^ 



It remains to reduce the last two of (6) to elliptic integrals. 

 If Jc and h' be the modulus and its complement, the following are 

 known transformations in the theory of elliptic functions, viz. : — ■ 



Hence 



W 7 



1 f V l-A 2 1 sin 7 cos 7 1 . 



r?7 1 & 2 sin 7 cos 7 

 A» = P PS ' 



tan 2 7 7 A tan 7 1 



