144 Astronomical and Nautical Collections. 



true sum 1.0460; if?-' = .43, 1.0218, whence / == 4210, 

 and r appears to be 37' 29". 



But if we wish to supply any real or imaginary deficiency of 

 the inverse series, we may easily revert to a modification of the 

 original solution of Taylor, who first applied to the problem of 

 atmospherical refraction his very useful theorem for " integra- 

 tion by parts," as the process is sometimes now called, that is, 



fZdY^—fZdX-d^L J*ZdX°~ + ...; and not 

 dX dX" 



fYdZ = . . . , as it has been inaccurately copied in the Article 



Fluents of the Supplement of the Encyclopedia Britannica, 



n. 5, 546. Taking the fundamental equation for the refraction, 



dr = — , and making first Z — — , dX = vdv, and dl" = dz, 

 v v 



we have for fZdx,f-Zdx' i , . . . , v, — v 3 , i> 5 , . . . , and 



O O . u 



for ,..., — l-,d — — : vdt>,.'..; or secondly, making 



dX vdv vdv 



dz ,z . zdv , (*dz z , fzdv 



— == d — + , we have / — s= — -f / , in 



v v vv J v v J vv 



which making dX again = vdv, and Z = z, d Ybeing = — , we 



vv 



have / — = — + — fzvdv + — f-zvdv.vdv + —!—..., 

 J v v v 3 v 5 v 1 



Now in all cases v- — x" — ti^, and vdv ss xdx — uda =.rd.r 



+ psudz, and since dx s= ~ -* = -Z_S — , we have = 



mz mz dz 



Z$ + psu = m P suz ~ <, and -±- = 2! 



mz mz vdv mpsnz — x£ , 



whence r = C^L = vmz - v " d vmz .1. 



J v mpsuz — x£ 3 mpsuz — x£ dz 



+ . . . j and from one or the other of the series 



mpsuz — xQ 



thus obtained, we may always compute the value of r, taking 

 the fluents from z ~ 1 to z c= 0. But at the horizon, it will be 



