356 Astronomical ariH Nautical CoHectioiis. 



now be convenient to divide ^ " into tlie two portions ^'\ and 



^" ,^ v', in order to obtain that part of the sixth term which is 



d'l) t" 

 independent of u ; the fourth will then become — :^ + 



dr' mpsz 



A . (-1 s\ + (-^ + ^^', + ^^ W . The whole 



mp's^z' \ mpsz j \mpsz vmp's^z^ mp's^zy 



of the fluxion of the former part will contain v, which will dis- 

 appear again in the next term, being changed into dv, and the 



v^ of the second part will become 2 — iu the sixth term. We 



dr- 



shall therefore have, for the case of the horizontal refraction, 



, ^ A ^ ^'" Mr, r, d2 r dv 2fdz 



when := 1 and s= 1, — = I — ^^-i — i-'. — _|. ^ - 



dr^ \f7ipd7- mp dr mp" d?- mp^dr 



^ + X . ^\ -^ + 2 ^' (Xi-+ 1.C+ ii-l. 



dr mp^ di'^\vdr dr''\mp imp" mp^ j 



1, • 1 • .u . • ddv ft' \ ? \ ,, 



It is obvious, that since ;=; — . j- — ^ — | v, the quan- 



d?" I V mpsz mp's^zj 



tity C must be derived from it by taking the fluxion with 



respect to v only, and must be equal to , which is the 



dr' vdr 



product of the second and third coefficients. The fluxion of 



this quantity, d^'\ is also capable of a simpler expression ; for 



d'^' dv ^' 

 since t will in general be divisible by v, t' =: -; — • i — = — 

 ' ° ■^ ' dv dr V 



^JL, and dlj^- ^:^; whence Xl =. ^. ^ _ il. ^^ 

 dr V V vv dr tjdr dr vv dr' 



^' d=u _ r ^ f " dt> X ^" - r'„^' ^ C d"u _ 

 V dr^ V dr v dr v di'^ u dr u dr' "~ 



y„ dv t d'v ^ ,, 



Q ^ V. \. Consequently 



dr V dr^ 



^fr_^ du _^ (pu _r_ di^_^_r_ ^ 2^ , 



dr* \mp dr v'mp dr' vmp' dr ' vmp' dr mp^ 

 dr J_ ddtjx dv_ dv^ (IC,, 4^' 4^ 



dr vmp' dry dr dr^ (^ mp vmp' mp^ 



ddu 



dr"" ^ ivp vmp' mp^ j dr \v pj vmpdr 

 6. We may next proceed to substitute, in these general ex- 



