128 TJ. S. COAST AND GEODETIC SURVEY. 



If we also wish to develop the Equator with the true 

 length, we should have in figure 31 0G = \, and, since the 

 angle 0P6^ is equal to the half of X', the triangle OPG 

 will give in turn 



, X' 2X 

 tauTT = — • 



From these two equations we obtain 



and also 



^ e 4\<p 



tan n — — V ' 



2 TT^ 



d(p ' _ sin (p' 

 dip cp 



d\' _ ^\n\' 

 d\ X ' 



so that we obtain 



, TT sin ^ _ TT sin (p' 



°" ~ 2 V? sin X' ~ 2 ^(1 + cos X' cos ip') 



, _7r sin B ir sin X' 



2 X cos (p tan <^' 2 X(l +cos X' cos (p') 



At the intersection of the Equator and the principal 

 meridian, we have 



^'m = 2 



fc'p = l 



Z' = 2. 



The Equator being developed with its true length, if 

 we make the second condition no longer apply to the 

 central meridian, but to the principal meridian, and if 

 we wish that the arcs of this last have for projections 

 arcs that are proportional to them, the relation between 

 X and X' will remain the same, but that which exists 

 between <p and (p' will be replaced by q>' = (p, which rela- 

 tions give 



, X' 2X 

 taUo = — 



Z IT 



. ^ 2X, (p 

 tan o= — tafirt* 



