and on Astronomical Refractions, 495 



Since 



u W = X 



I 



d x , ado) , a 

 d m i d u i 



dw_dw d x 

 d u ~ d x d n 



therefore 



d (o 

 dl 



1 r Wi 



I 



similarly 



d 2 u> (i) 2 



d.l 2 



d 3 w _ w 3 , 



d~T 3 "~ "• * \ * ~ t " 



3 — K) 2 



10 -7- W 3 W 2 l«* "«f ( w 2) 3 



d j:« - « T - ■ ^ 



o-f-y o.-f-y (f-T-y 



The quantities y^, r r^2 5 &c, might also be deduced from 

 -^Yp -T-yj~2> &c., by similar expressions, only changing the signs 



of those terms which are multiplied by uneven powers of — . I 



have not, however, found it convenient to have recourse to this 

 method of obtaining the development of w in terms of X. I have 

 employed the series 



, ' v a d(FX) 2 a 2 d 2 (FX) 3 1 



9 

 fit (A, 



and I have found — . (F Xf by actual multiplication, . 2 (F^) 



by multiplying^, (FP) by^, (F X), &c. This process, though 



somewhat tedious, is extremely easy. As it may be carried on sy- 

 stematically, and the numbers follow each other, it is not liable to 

 error. 



So far all is general ; it now remains to make some supposition 

 with regard to the function f u, upon which the constitution of the 

 atmosphere depends. If we take, as in vol. xvi., p. 440, see also 

 pres. vol. p. 279. 



*= i - H^-y c ~ u \c~ u - i + HJ y~ l 



