140 TJ. S. COAST AND GEODETIC SURVEY. 



intersects the parallel, and let S be the center for the latter; 

 draw the abscissa MN of the point M and the tangent MT 

 to the ellipse; also draw >S'Z7 and SM, 



The parallels are the same as those in the globular pro- 

 jection, so that we have, as before, 



s — r=<p 

 7-2 = §2 + -— — tts sm ^ 

 or, by combining the two equations, 



<p(r + s) —tts sm. <p+-T = Q 



4-^ 



TT sin (p — 2(p 



'Bj taking the derivatives of the two members of these 

 equations with respect to <p. we obtaiu 



ds 2r — Trs cos (p 



d<p IT sin (p — 2(p 



d(p d<p 



The angle OSM is still denoted by 6. The triangle SMN 

 gives for the rectangular coordinates of M with as an 

 origia 



x = r sin 6 

 y = s — r cos d. 

 The elliptic meridian has the equation 



.(*)•=:. 



By substituting the above values of x and y in this equa- 

 tion, and then solving for cos 6, we find 



^ X V4X* + 27rX2 (2s sin ip-ir)-\- ttV^ - 4X^5. 

 cos e= r(7P-4V) 



