102 U. S. COAST AND GEODETIC SURVEY. 



We thus have OGxOG' = 1, which ought to be so, since 

 OP is a mean proportional between 06^ and OG' . 



The coordin:ites <^' and X' or p' and X' determine the 

 position of any point of the map; however, we shall make 

 use also of a third variable depending upon the first two. 

 This will be the angle OSM formed by the radius SM of 

 the projection of the parallel with the straight line meridian 

 or, what amounts to the same thing, the angle OTM 

 formed by the radius TM of the projection of the meridian 

 with the straight Hne parallel. We denote this angle by 

 0; it is the angle at which -one would see, either from the 

 center of the projection of a parallel or from the center 

 of the projection of the meridian, the distance of any 

 point M to the center of the map. 



Half of ^ is equal to the inscribed angle OG'M, which 

 subtends upon the circumference T the same arc as the 

 angle at the center OTM, or to the angle OG'D, since 

 the three points G' , D, M are in a straight line; but the 

 tangent of this angle is given by the ratio of OD to 0G\ 

 We have, then, 



tan ^ = tan -^ tan ^ • 



From this equation we deduce 



2 tan -pr • N / • r 



. 2 _ sm X sm (p 



sm d= - — — ^ - i+cosX'cos<i?' 

 l + tan^- 



1 - tan^ 



COS0 



1 — tan2_ - , , , 



2 cos X +COS <^ 



^ , B 1+cosX' COS ip' 

 l + tan^^ 



The coordinates" of M with respect to the axes OA and 

 OP are 



. ^ sin X^ cos V?' 

 a; = rsm0 = 



1+COS X' cos (p' 



y=^ ^^ ^= i+cosy— -' ■ 



sm (p_ 



cos (p' 



