and on Astronomical Refractions. 

 2 a sin OA e c"* "* x /' e * ( * " * 2) tf" d j? 



493 



V 2i x n 



which is to be integrated from z = to z== 1. This integral may 

 be exhibited under two aspects; in the first, which is that given by 

 Mr. Ivory, the coefficients of the different powers of e consist of a 

 finite number of terms. In other forms of the integral, which will 

 be given here, applicable to all atmospheres of finite altitude, the 

 coefficients are composed of an infinite number of terms, conver- 

 ging with rapidity and in a form suited for numerical computation. 



1.2 



-b 



and the single term Ac d x in d oo will give in 



/ 



a (1 + a) sin 6 doo 



\/ cos' 2 6 + 2 ice 



the term 



2A*(l + 



$^{{"- 



b 2 x" 2 



+ *(1-*) lx"*Z-bx"*Z> + 



63 a V/3 

 63 jp//5 



Z 3 + 



&C.| 



—2 — TT^+^r d 



62^/5^4 63^/6 -| 



~2 273- Z+&C 'i e d * 



+ &c. 



} 



v ; (m + 1 Wot + 



2) (»- 3) 1 



(ot + 1) (ot + 2) (m + n + 1)' 



Hence it will be seen that if d w contains any number of terms 

 of the form A c~ x d x, the definite integral required 



2 (!+«)« sin 4 J r /VdwV, *"3 /-dw 2 V, a"* /d3 w \' ") 



= ^27^ [{* + x(W+2T3Vd^)+2T3T4(d^)+ &C -} e 



+ l2.3Vdx/ + 3.4Vd*V +2.4.5 Vd *V + J ' ^J 



+ isTO Vd^y + 4T5T6 (d7O+2.5.(i.7(dT0 + &C 'f e 



+ &c. I 



Another expression for 8 6 may be obtained in the following 

 manner. Suppose d w contains any term 



A n (x»-x) n &x = A n X n &x 



