324 Prof. G. H. Darwia. On Jacobs Figure of [Nov. 25, 



Approximate Solutions of the Problem — continued. 



Now, since (16) may be written 



1 -™ & 2 si n 7 cos 7 & 4 . „ „ T 1 ^ A, ~I 

 - (2F - E) Pf-^ = T2 sm^ 7 cos^ ^ - - E + - tan 7J , 



it follows from {a) and (8) that it may be written 



^Z^ = sin^cos^ri^d 7 . 

 A3 Jo a 



A- 



Again, we may write (19) thus : — 



_w2 ^ A co s 7 _ Acos 3 7 E _ A 2 cos 2 7 

 47r ^ 2 sin 3 7 &' 2 sin 3 7 &' 2 sin 2 7 



A cos 7 1 sm2 7^ A cos 3 7 f tan 2 7 ^ 

 ~ sin 3 7 J o A 7 sin 3 7 J Q A 



A cos 7 f" [ y sin 2 7 , „ f Y tan 2 7 , ~| 



It is easier to develop the equations when written in the forms (J) and (c) than 

 when we work directly from the elliptic integrals. 



Write for brevity 



& = sina, ^^cosa, _p = cos 7 , g = sin7, Q,=tan7, A = log e cot(57r — £7). 

 First, when k is small — 



The following definite integrals are required : — 



(d) 



1 1 3 3 5 



From (d) and from — = (1 - Pg2)f = 1 + g *Y + ^ V + . . • 



q* 1 , 3 3.1 3 , 2 / 1 , 5 a 5.3 5.3.1 \ 



V_dy=- p >p-- qP + -y + -^--^ 



A* = l-k*q 2 , and 



o< 1 . 3 3.1 7 T /3.1 l\ , /3.5 3.l\ _ 



3.5.3 /3.5.3.1 3.1 _\~| 



1 , 3 3 79 f 1 . 15 /15 3 2 \"1 



Again, — *= 1 + ^-fc 2 ? 2 + . . ., and by (e) — 

 A A 



