Mr. Ivory on the Theory of the Astronomical Refractions. 281 



. , • u 1- • • l^B 



or, which IS the same thing, putting i = „ , 



sin 9 



sin d) = :; r . 



1+i 

 Again, i being the angle in which the light of the star is re- 

 fracted, if we put S9 for the refraction, the angle of incidence 

 S B H, which in the present case is always greater than the 



angle of refraction, will be = ($ + S 9 ; and ~ ^will 



° sin 4> 



be a constant ratio represented by =y^; so that 



sin(<f) + 8fl) 



v'l— 2a 

 sin <p sin 



-v/l-Sa (1+0 v/l-2a 



Thus we have the two following equations, which furnish a 



very easy rule for computing the mean refractions according 



to Cassini's method, viz. 



. ^ sin . 9 

 sm $ = 



sin (^ + 8 9) = 



1+? 



sin 9 



(l+tVi_2a* 

 As i and a are both very small numbers, if we put 

 m = i — i^y 



So." 



n=t — u — r + ut ^p- , 



the two last equations will become 



sin ^ = sin 9 — m sin 9, 



sin ($ + S 9) = sin 9 — 71 sin 9 : 



and by employing the usual formula for deducing the varia- 

 tion of the arc from the variation of the sine, we get 



»?=' 



consequently 



<f = 9 — »j tan 9 + — tan^ 9, 



<f + 8 9 = 9 - n tan 9 + -?r tan^^: 

 2 



8 9 = (m — n) tan 9 . tan^ 9 ; 



that is, 



3« 



89 = (a -/•« + ^^^) tan9- ( ««--^) tan39; 



or, which is the same thing, 



89 =z a tan 9 (l +a- '^~) = «(l +«) tan9 ( 1 - ''{ *), 

 ^ cos** 9 ' ^ ' \ cos^ 9 ' ' 



