Mr. Ivory on the Theory of the Astronomical Refractions. 501 

 The coefficient Agn + i is thus expressed in terms of j;: 



the indefinite integral is 



r,,, , , d.P{x) . dd.P(x) „ 1 



This integral, taken between the limits o^ = and x — ;«, is 

 equal to Agw+i : the first form of *•' {x) will give the values of 

 all the differentials at the limit x = 0\ and the second form 

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

 X — HI'. Thus we obtain, 



. r. w + 1 , n-l n+l.n + 2 - ") 



■"^ \_- m 2 m^ J 



— «. r, . '^+1 n—l n+l.n-i-2 „ ] 

 + €-"". < l-{-n. +«--^7- • ^-^ + &c. y, 



the upper or lower sign taking place according as fi is even 

 or odd. 



The numerical coefficients, computed by the formula, are 

 as follows : 



c-*" = c-^** = -0000454 



Ai = 1 — c-"* = 0*9999546 

 4 6 

 T+ 5 



Ag = — + —c-^ = 0-8000545 



^5 =^^^5-^^'^-'"= 0-5199219 



A, = ^ +^c-'« = 0-2801326 

 ' 25 25 



16 726 



31 8359 

 A" = 625 + -625 ^-"" ''■«^»2''^2 

 Ai3 = 0-0172805 

 A,5 = 0-0052779 

 Aj7 = 0-0014467 

 Ajg = 0-0003593 

 Ag, = 0-0000815 

 A23 = 0-0000170 

 A2.5 = 0-0000036. 



