MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 205 



and, by performing the differential operations, 



^ W = x(i - c-0 +/(Ri + R2) +/(R2 + 3 R3 + R4) ; 

 and, by substituting the values of the functions, 



A = 2/— X = -22566 



lf(x)=:-h{\-C-^) -i-fx + 4/ (1 - ^ + ^' - f^ - C- '^). 



It might not be very objectionable to neglect the term multiplied by/', for the same 

 reasons that the terms which follow it are neglected, that is, both on account of the 

 nature of the functions and because the coefficients are small : but, in order to leave 

 no room for scruples respecting accuracy, the square of the entire expression set down, 

 may be thus represented : 



Y2 (x) = G - 8 A/ . G' + 8// . G" + 1 6/2 . G"'. 



The integral in the term under consideration is greatest when the radical quantity 

 in the denominator is least, that is, when cos ^ = : and if the integration be per- 

 formed betwen the limits x = 0, x = 03, we shall obtain a result greater than if the 

 integral were extended only to the top of the atmosphere. Now we have, 



G = A2(1 -2c-' + c-''')-^2hf.xc-'-2hf.x+f^.x^: 



and, by operating on the terms separately, the part of the integral depending on G, 

 will be as follows : 



^ 



dx dd.c-'^G 



V%ix dx^ 



-^. X (a2 (1 - 4 v^~2 4- 3 ^"3) - 3 hf{s/^^ 1) + \P) = ^. X -00216. 



The other parts depending on G', G", G'" are complicated ; but they are troublesome 

 more on account of the number of terms they contain than from any difficulty in the 

 integrations. The following results have been obtained : 



r dx dd.c~^G' ju a/Z 



r^ dx dd.c-'^G" _ ^ x/;r 



Xc/o ./7n~. ' ^^2 — ""V X — T== 



S/ZX^o ^T^T^- '^^^ =-/x -^X 02043, 



16/2^ -^ ' "^^'"'l^" = +/2 X -^. X -00855. 



Collecting all the parts, the term sought is found, viz. 



«( +«) r'^ dx d d . c- "= "if^ (x) _ 

 2 '*^o v^an- * dx^ "~ 



a (1 + a) V^TT 



>/a 



X (-00108 -/ X -00142 +/'2 X •00427). 



