134 IT. S. COAST AND GEODETIC SURVEY. 



expression for ^m, p by X, r by R, s by S, and 6 by 5, and by 

 dividing by sin p; this gives 



, /dR dS A 



sec \f/ 



sin p 



The projection of TIT upon Or being equal to TO plus the 

 projection of SM, we have 



i? cos S^/S' + r sin ^. 



Substituting for cos 5, in the expression of Icp, the value 

 which results from this last equation, and observing that 



B-nT- — S-3r- is zero, since R^ — S^ is a constant, we have 



but 



so that 



, r sin 6 dS^ 



^~ R sin p cos \f/ d\ 



ldS_ l_d}^ 



R d\ sin X' d\ ' 



T _r sin 6 sec \l/ dX' 



P sin X' sin p d\ 

 The expression for Z^m can be written 



^- = [^^-2 ^ sin^ I] sec ^. 



Let us examine in particular what these ratios become 

 upon the straight-line parallel of the map which we shall 

 make, for example, correspond to the Equator. Let us 

 call A the value which is assumed for ^ = by the deriva- 

 tive of OD or s — r with respect to (p and —B the limit 



. ds 



toward which tends the ratio of -j- to 2r^ when <p tends 



toward zero. Since at the same time rd tends toward OG 



or tan -^ , we find that on the Equator 



y 



2 



lcm= A-^ B tani^ ^ 



1 2 ^' ^^' 



^p==7^ sec^ -p^ -j^7 



since ^ = at that point. 



