THEORY OF POLYCONIC PROJfTCTIONS. 141 



By using this equation we can compute the angle d as well 

 as the values of x and y. If we denote by 77 the angle OTM 

 formed by the tangent to the eUipse at M and the Y axis, 

 we know that we have 



4X2y 

 ^^^'^^^^^ 



but the departure ^ of the angle of the meridian from an 

 orthogonal intersection with the parallel is the angle SMT, 

 which is equal to the difference between the angles OTM 

 and OSM; we have then 



Everything is now known in the expression for Icjr^, namely 



By substituting the values this becomes 



7 /-I , r.'^s cos (p — 2r . ^e\ - 

 fCm=i 1 + 2 -• — o sm^ ) sec \pr 



an expression that has the same form as in the case of the 

 globular projection; but, of course, the angles 6 and ^ have 

 different values from what they had in that projection. 



K=^^^^<^-p- 



By differentiating the equation for cos 6 with respect to \ 

 we obtain the value of ^ which may be reduced to a con- 

 venient form by substituting for sin its value in terms of 

 X and y; this form is much more readily obtained by dif- 

 ferentiating the expressions for x and y with respect to 

 X, and then the differentiation of the equation of the ellipse 

 partially with respect to X will furnish the equation for 



determining ^' In this way we get 



^-r cos e^=is-y)^ 

 by • ^be b9 

 aad 



X^dX X«"^7r2dX~"- 



