212 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



If we take the integral between the limits x = 0, x = oo, the result is not sensibly 

 different, viz. 



a (1 + a) / ,-—r~ = -^^ — 7= = 2036"-52. 



■ Investigation of X X Qi. ^ 



For this purpose we must find the value of 





^(,_,.). + 4e^-^ -^o y/(,_^. + 4.-|£ 



2 w 



this integral has therefore the same form as Qo, the quantities 2 x and 2 m taking the 

 place of X and m. Wherefore, if we assume 



^ ^S — ^^ = ai e + ag e3 + as e5 + &c. , 



the value of ag „ ^ i will be found merely by writing 2 m for m in the expression of Ag^ + 1 ; 

 but as c"^*" = c~^° is extremely minute, the part multiplied by it may be neglected. 

 Thus, 



The numerical coefficients are as follows : 



ai = 1 



a3 = 0-9 



as =073 



ay = 0-536 



ag = 0-3555 



an = 0-21505 



ai3 = 0-118945 



ai5 = 0604215 



ai7= 0-0283127 



ai9 = 0-0122898 



agi = 0-0049621 



a23 = 0-0018695 



a25 = 0-0006623. 

 These values being found, if we assume 



X X Qi = B3 e3 + B5 e5 + B7 e7 + &c., 

 the term X X c~ "* X e, which is insensible, being omitted, we shall have 



B3 = X (ag — A3) = 0-021866 



I 



