THEORY OF POLYCONIC PROJECTIONS. 151 



SO that 



d8=a? [cosec^ <^ — cot^ (p cos (X sin ^)] cos (p d(p d\. 



One-fourth of the area is given by integrating between the 

 limits X = to X = 7r and <p = to <p = 2' The total area S is 

 therefore given by the formula 



S = 4ja^ I ^ cos <p d(p I [cosec^ ^ — cot^ (p cos(X sin <p)] d\ 



Jo Jo 



= 4a^ MM" cosec^ <p — • 3 sin (t sin (p) cos <p d(p 



— I — COS^ iD » 



= 4a^ [ — TT cosec <p] 2 —4:0^ | 2 . sin (ir sin c>) (Z^. 



In the latter integral let « = tt sin <p 



then 



J dx 

 cos (p a(p= — t 



TT 



and 



/'^ COS^ 



— 4a^ I 2 . „ sin(7r sin <^) cos d(p 

 Jo sm^ (p ^ 



i 



Aim T^ ■ ^^ 



X^ TT 



- 2 , ri sin a; , 1 cos xT , /^ 2 . ^n 2 T'^sin a: ■, 



Hence the value of S becomes 



/S^ = 4a2[-^ cosec d^ + 27rVr^+^^^T 



o \_ Xr X _\o 



Jo ^ 



The integrated terms assume the form 00 — 00 at the lower 

 limit, and must be evaluated for that point. The last term 



