MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 211 



The coefl5cient A2„^ j is thus expressed in terms of x : 



the indefinite integral is 



This integral, taken between the limits x = and a: = m, is equal to Ag^^ i : the 

 first form of T"' {x) will give the values of all the differentials at the limit jc = ; and 

 the second form of the same function will give the like values at the other limit 

 X =■ m: Thus we obtain, 



A2„+i|l ^ m ^^ ^Z m^ -&C. j- 



the upper or lower sign taking place according as n is even or odd. 

 The numerical coeflScients, computed by the formula, are as follows : 



0-" = c-^° = -0000454 

 Ai = 1 — c-"' = 0-9999546 



Ao = 4 + 4 c- *" = 0-8000545 



13 43 



7 73 



A, = ^ + ^ c-"* = 0-2801326 



Ifi 72fi 



A, =jg5- {53 c-» = 0-1277363 



A21 = 0-0000815 



A23 = 0-0000170 



A25 = 0-0000036. 



The horizontal refraction answers to cos ^ = 0, e = 1 ; and the part of it depending 

 on Qo is found by adding all the coefficients, viz. 



"* ^\— "^ X 2-8024736 = 2036"-52. 



V 5i 



2 E 2 



