22 U. S. COAST AND GEODETIC SURVEY. 



If the whole area of the sphere is represented on one con- 

 tmuous map, one-fourth of the area of the representation 

 will be given by integration of this expression from X = 



to X = 7r and from <p = to <^=-^. 



To obviate the use of the fractions, it is better to let X = 2y; 

 y will then range from to ^ and (ZX = 2 dy. 



The total area S is given by 



+ cosec^ if tan -^ l-^sin if y 



cy A o C-V TT COt^ (p cos ^ , o ? X 



^ = 4^2 I 2 ^ ^ + 2 cosec V cot (p 



(l+^sinV) 



tan -M 2 sin (y? j d(p. 

 >S' = 4a2 ^ cQsec (^ — cosec^ ^ tan "M ^ sin <p j U 



(s-) 



tan -' I 



The quantity in brackets has to be evaluated for the lower 

 limit, since it takes the form oo — oo at this point. Let us 

 write it in the form 



^ sin (^ — tan ~M :o sin <^ j 



sm^ ip 



which takes the form jr at the lower limit. 



<p^^OL sin^ (f _J 



