90 



Hypothesis of uniform density. 



5 Let CR be the radius of the uniform atmosphere, the 

 height of which is / (vid. Fig.) 



^ = angle of incidence at the point R; t = VRC^ then 



ref. iR') =^ O' — t, and -~ sin. 6 = sin. t = ^^- (1) 



Hence am sin. ^ = (a+0 sin. (/ + E) (2) 



but supposing the surface of the earth a plane 



ms'm.6 = sin. (^ + Q) (3) 



Hence sin. ^ + E) = ""' ^'+J^ (4) 



a 



making Z, ^ and R to vary, in order to apply Taylor's 



Theorem. 



'^ ' Byequat. (4) 



O+R) cos. (t+R) = - _ ~/ I ^» sin. (^+Q) 



By equat. (I) 



f G0S.4 ^ ""' i ' . sin. ^ 



a(l+__) 



Hence computing R+R + 8cc. making R= Q, t==6 -^o 

 and then — = -^, we have by Taylor's Theorem 



R = Q _ I- ( tan. (^+Q) — tan. ^ ) + &c. (5) 



But tan. (^ + Q) = tan. & + -^^ + &c. 



