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



put the integral Qg, taken indefinitely, in the following form, 

 which it is not difficult to verify by differentiating : 



*e .Q.dxc-^'^ 91 pedxc-"" 



Q3=_4y _^ — +167- 



125 ^\ pedxc—'' 



/21s . 175 , 125 „\ /> 

 ^ V 16 ^ 16 48 / \y 

 c-^A /185 125 o 125 ,\ 



V16 ^ 12 ^48 ^ 



c-^A/95 5 „ 25 



e V24" 12"" ■ 24^'^ )' 

 This being the indefinite integral, the value of Qg in the 

 formula for the refractions will be obtained by putting x = m 

 = 10 ; which gives 



e e ' ' 



and this value, as well as that of e, being substituted, the 

 quantity sought will be expressed as follows : 



^ . re.'ldxc--"' 91 pedxc-"" 



+ 



A 



^edwc~ 



\2i5f\-e^Y 175/ l-g^ \^ 125/ 1 — g^ \«1 fe 



\ 16 V^) "^ T6'\~7~'y' "^"48 VT";' ]^ ~ 



f\25 i 125 1 905 1 905 125 o 127 ,\ 



I ^-m I J |_ p pS I pb I 



^^ U8 g^ 16 g ^ 48 e ^ 48 16 ^ 48 7 



~'^*?~'^ 48^^^" 48 g 48^ 48 ^ "^ 1^^ • 



The series equivalent to the integrals must now be substi- 

 tuted, in order to express the quantity sought in terms con- 

 taining the powers of e. 



In the first place we have these three terms, each of which 

 is zero when the exact values of Aj, Ag, &c. are substituted, 

 viz, 



48 ^ ^ ^ e 



{115 . 125, . ^ . , 125 125 ^1 J 



