THEORY OF POLYCONIC PROJECTIONS. 131 



sin p'' 

 sm € 



sin 7 



s = —. — - 



sm € 



cos (r^) 



e 

 COS -^ 



sm 



^i^) 



sin I 



The ratio of the two parts DP and PP' into which the 

 line PP^ is divided by the projection of the parallel is 

 expressed very simply by means of v' and y. In fact, 

 this latter angle is equal to that of the two tangents at 

 U to the two circumferences, which angle is divided into 

 two parts by the chord UU^, the one of which is the double 

 of the angle DUU^ and the other of the angle PW . 



The angle PJJB is then equal to ^, but of the two comple- 

 mentary angles PP'U and P'PU the first is equal to ~ . 

 It comes about, then, in the triangles DPU and DP^U 

 that 



Pi7sin| = PPcos| 



P?7 cos I = Z>P' sin I', 



from which, by dividing member by member and on 

 denoting the ratio by ^, 



DP . ^ v\ ^y 

 ■pp7 = ^ = tan| tan^- 



