MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 203 



it may be taken equal to a, or to the mean value a (1 + «)• Thus we have 



TV.. • A u(l + a)duc-'^ 



cos^d + 2i- -f Sil 



Again, the formula (9.) gives 



' ? \ '' c~^du '' c-^.du^ J 



Now, 



p!_ y 



and if we make 



pT(r^-(l+/3r') = L(l+.3r'): 



^- p' ■"L(l +/3t^) — *^^ 



. /, -MX /. <^-g "1^2 j^dd.c "R4 



^ ^ "^ c-'^du '' c--^du^ 



Let "SP" (m) stand for all the terms in this value of x except the first, so that 



X =^ u — '^ {u)'. 



from this we deduce by Lagrange's theorem, 



c =c -c^^(^)--2 ^^^^ -&C.: 



consequently, 



due ""—dxc A -i — ^—^dx-{--^' ^-5 — —^dx-{-hc. 



' ax ' 2 da;'' ' 



By means of the values that have been found, the differential of the refraction can be 

 expressed in terms of one variable x. In making the substitutions, the smallest term 

 of the radical quantity is to be neglected in all the terms of du c~'^, except the first 

 and greatest ; and the denominator of that term is to be expanded. Thus we obtain 



^ * ^ l^ x^cos^Q + Qtx \ ' dx J 



, 1 / dx dd.c-'"¥^{x) 



+"2 y 



2 J 



\^cos^Q + 2ix dx 



dx .c~'' .i^x^ 



(cos^e + 2ix)§ 



In order to estimate the relative magnitude of the several parts of this formula we 



2 D 2 



