and on Astronomical Refractions. 495 



Since 



a. dx.udai.ee 



U r- W = X -5 = 1 -r- j = 1 ' -r- Wj 



I du I d U 1 



do> _ d w da; 

 du ~ d x dw 



therefore 



d w _ fa>i 

 d^ = ' 



1 Wj 



similarly 



3-fl ( w ) 

 d 3 0> Wg I 



4 



d 4 <" <d< I . 



d.r< /, \ 6 /, * \ 6 /, * V 



(i-jfi) (i-T"") s } -Jy 



The quantities y-^L j^> [&c., might also be deduced from 

 TT7 T772 ^ ClJ ky similar expressions, only changing the signs 



of those terms which are multiplied by uneven powers of -r-. 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 



p v" a d (FA.) d (r .X) 



w = 1 r .A . , -.r ~~ ^ ;, -o i ^75 occ., 



and I have found -. (F X) 8 by actual multiplication, 



2 



* o Z 



by multiplying^-. (FX 2 ) 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. 





