THEORY OF POLYCONIC PROJECTIONS. 



23 



lim 



COS ip — 



2 cos<^ 



l+-T-sin2 ^ 



lim 



Therefore, 



2 sm (p cos <^ 



<^ = 



7r« . 



16 



sin (p 



1 + 2" sin^ (^ 



= 0. 



/S' = a2[(4 + 7r2) tan-i | + 27r]. 



This value is greater than the surface of the sphere in the 

 approximate ratio of 8 : 5. 



The length of the outer meridian for the representation 

 of the sphere is given by four times the integral of a Icjq, dip 



from ip = Q to ip = ^ with X = r in the value of d. 



For the sphere Tcj^ — cosec^ ^ — cot^ tp cos d, 

 and for the outer meridian 



tCm, 



1+^ (1+ COS^^) 



1 + J- sm^ (p 



The length of the meridian is, therefore, given by 



^1+^(1+ cos^'^) 



2 _ ^^^ 



l+^sin2 ^ 



By means of a table of integrals we find that the value of 

 this integral is given in the form 



Z = 2a7r[(4 + 7r2)J^2_l]. 



The length of a great circle at the outer limit of the map 

 is increased in the ratio 



(4 + 7r2)H -1 : 1 or about 2.72 : 1. 



