132 U. S. COAST AND GEODETIC SURVET. 



The alteration \f/ of the angle of the meridians with the 



parallels is the excess of the angle SMT oyqv o- In order 



to obtain it simply, let us note that, M^ being the second 

 point of intersection of SM with the circumference PMF\ 

 we have 



SMxSM,=:SPxSP\ 



if If is displaced by changing the meridian but, remaining 

 on the same parallel, SMis constant; then the same is true 

 oi SMi; consequently, also of MM^. Then the projection 

 MN of the radius TM of the variable meridian of the map 

 upon the radius SM oi the fixed parallel has a constant 

 length. At the point M this length is expressed by i? sin ^ 



or by - — ^1 and, at the point U, by cos 7; it thus results 

 sm A 



that 



sin ^ = cos 7 sin X'. 



In the triangle OST the angle at S^ which we will call <r, 

 may be immediately obtained, for we have 



tan 0- = — 

 s 



Let us now designate by 6 the angle OSM and by 5 the angle 

 OTM, which we shall need for calculating the ratios Arm 

 and fcp. The triangle S TM gives 



■p 



sin (^ + 0-) = -jTo cos ^ 



cos (5 + <r) = -^ cos \p; 



but we have in the triangle OS T 



S s 



S] 



so that we have 



TS^-^ , 



sm 0- cos a 



T) 



sin {d + cr) = -^ sin <r cos ^ 



v 

 cos (5 + 0-) =- cos a cos ^ 



