Mr. Ivory on the Theory/ of the Astronomical Refractions. 505 

 and we shall find 



5 5 



Can+i = Sasji+i rpAsM+i ^A^A2,i_], 



The numerical coefficients will now be obtained : 



A^Ai =—'0802325 



A^Ag =+-04034.33 C5 = '059337 



A^Ag = '0873930 C7 = -151186 



A2A7 = '0748672 Cg = -204491 



A^Ag = '0446024 C,i = '193076 



A2Aii= -0209241 C]3= '142381 



A^AigS '0081714 Ci5= '087220 



A2Ai5= -0027438 C^^ = '046149 



A2Ai7= '0008096 Cj9= -021658 



A2Ai9= -0002133 021= -009187 



A^A2i= '0000511 €33= -003569 



A^A23= '0000105 €35= -001290. 



As the value of / is not fixed with the same certainty as 

 that of A, the coefficients of Qg have not been multiplied by 

 /: the intention of which is to make it more easy to deter- 

 mine a variation of the refraction, viz. If x Qgj answering to 

 Syany variation of y that good observations may require. 



The part of the horizontal refraction depending on Q^ is 



-I X "^^-tJ^x 0-919534 = 148"-51. 



If we integrate the original expression of Q^ from .r = to 

 .r = cvD, e being 1, we shall have 



-Xal+«)x/ ^ /cT^ ■ 



= 1. «ii + «IV;5 . (2 ^2 - #) = 148"'63. 



It thus appears that the error is less than 0"-12; for the exact 

 integral from x = to ^' = w = 10 is less than the second 

 number, and greater than the first on account of the terms 

 of the series left out. 



The next point that should engage attention is to find the 

 value of/' X Q3. In the present state of our knowledge 



