The Variation in Attraction Due to the Attracting Bodies. 223 



By construction angles RFC and E P C are equal, and either of these 

 angles is equal to half the difference of the angles P E D and E P D or 

 E P B. Angle C P G is also half the difference of angles P G F and G P F 

 or G P B. Diagonal P G is tlie direction of the resultant of the two forces 

 represented by the lines 1 and (1+a)-. 



F 



Angle R P E is equal to angle (u+v). 

 In triangle P E D, 



tanPA.^lL^Z^=tan CP E yi^. 

 2 a 



In triangle P G F, 



tan PG_F4^G_P_F ^ ^an C P G ^_±^^^^ 

 2 2 a + a'^ 



The half sum of angles PED and EPD is equal to the half sum of angles 

 PGF and GPF, therefore 



tan CPE 2+^ = tan CPG ^^^^ 

 a 2a+a- 



tan CPG 



2-ia-+Aa3-ia-' + „ 



tan CPE 

 From values of A„P and a,,,P already developed in this Art., 



a = 2\/1 +E- sin 3- cos 5 sin a— 2£V1 +^'" sin^ Sr cos 5 sin a + 



(1 + E-)- sin^ 5 cos^ sm^ ^ '^ + ., 

 a- = 4 sin- 5 cos- 3 sin- ti+-„ 

 The angle CPE is one half of angle RPE or i (u+v). 

 U4- V 



tan 



1 — cos (u + v) 1 1 



sin (u + v) = ~2' '^^ ^"+^> + 2A ^^^' (u + ^) + etc. 



tan CPG=2(1— sin- 5 cos- ^ sin- a + Jtan — g- 



