392 Mr. Ivory on the Theory of the Astronomical Refractions, 



and in parts of the radius, 



a. = -0002835. 

 It has been found that L = 4-34.7*8 fathoms at 0° centi- 

 grade or 32° Fahr. : wherefore, if we make a = mean radius 

 of the earth = 3481280 fathoms, we shall have at the tem- 

 perature of our climate, or 50° Fahrenheit, 



, = -i-^ = -^81280— = -0012958; 

 and hence ;, = 4 =-21878. 



We can now inquire into the values of the last two terms 

 of the foregoing formula for the refraction, both of which are 

 very small. With respect to the first of them, we have 



^ ^ ^ ^ "^ c-'^dx "^ c-'' dx^ 



and, by performing the differential operations, 



*(a:) = A(i-c-)+/(Ri + R2)+/'(R2+3R3+R4); 



and, by substituting the values of the functions, 

 h =2f—\= -22566 



!P(^)= -h{\-c—) + foc + ^f{\-x-V^-^-c-y 



It might not be very objectionable to neglect the term multi- 

 plied 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 : 



r-(^) = G-8 V. G'+8//'. G'' + 16/'^ G'". 



The integral in the term under consideration is greatest when 

 the radical quantity in the denominator is least, that is, when 

 cos fl = : and if the integration be performed between the 

 limits a; = 0, .r = oo, we shall obtain a result greater than 

 if the integral were extended only to the top of the atmo- 

 sphere. Now we have, 



G = A«(I - 2c-^ + c-«^) + 2^/.a?c-^-2 7//.^+/^.^^: 

 and, by operating on the terms separately, the part of the in- 

 tegral depending on G, will be as follows : 



»c^ dx dd.c-^G 



f 



V2ix' d,^^ 



