100 U. S. COAST AND GEODETIC SURVEY. 



that which may be made use of in the stereographic pro- 

 jection upon a meridian. 



Let us construct TL perpendicular to TP and inter- 

 secting in L the projection PMP' of the meridian; the 

 three points P' , L, A are in a straight line, for the angle 

 PP'Lj which has its vertex upon the circimif erence T and 

 intercepts the same arc as the angle at the center PTL, is 

 equal to half this angle or to half a right angle ; therefore, 

 the prolongation of P'L ought to pass through the point -4. 



The radius OP or OA of the circumference described 

 upon the line of poles as diameter being taken as unity, we 

 define the modified latitude of a parallel as the arc ^ Z7 of 

 this circumference comprised between the straight line 

 parallel AA^ of the map and the projection UDU of the 

 parallel in question. This arc which we denote by <^' is 

 also the half of the angle at which, from the center of the 

 projection of the parallel, one would see the circumference 

 described upon the Hne of poles as diameter; this arc varies 



with (p from to^ and from to — ^. For the abbrevia- 

 tion of the formulas we shall often use in them in place of 

 the arc that has just been defined the modified colatitude 

 p\ which is the complement of (p^ and which represents the 

 arc PU comprised between the projection of the pole and 

 that of the parallel; p^ can then vary from to tt with the 

 colatitude p. 



Every circumference described from a pouit S of the pro- 

 longation of PP' as center, with the tangent SU for radius, 

 is, in any system of projection with orthogonal intersec- 

 tions and with circular meridians, the projection of a par- 

 allel; that which varies from one system to another is the 

 position of this parallel upon the globe, or, inversely, it is 

 the expression oi v?' or of p^ as a function of cp or p, respec- 

 tively. Whatever this expression may be, if we call r the 

 radius SD or SU or SM of the projection of the parallel 

 and s the distance OS from its center to the center of the 

 map, we shall have from the right angled-triangle OSU 



r = cot (p^ 

 s = cosec (p' 



