218 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS, 



written, 



and the formula for the refractions will now be, 



\ »tf=sin(IX^^^'(Q„ + XQ,-/Q2-/Q3). 



Suppressing the tedious operations of reducing, we may put the integral Q3, taken 

 indefinitely, in the following form, which it is not difficult to verify by differentiating: 





e 



,— 2x 



^ , Pe.ldxc-^" . 91 Pedxc-^ 

 Q3 = - 4y -^ + Yd "~A~" 



, /215 „ , 175 . , 125 A redxi 



C2^/185 125 125 A 



^ e \ 16 ^^ 12 ^ ^^ 48 V 



/185 , i^5 125 A 



-^Vi6 +-r2^' + -48^7 



/95 A 2a.?^ 2^ 

 \24'^ " 12*^ "T-24'^-^ /• 



c-*A 



This being the indefinite integral, the value of Q3 in the formula for the refractions 

 will be obtained by putting a? = m = 10 ; which gives 



c-'A 1 + e^ 



c 



and this value, a'S well as that of g, being substituted, the quantity sought will be ex- 

 pressed as follows : 



^3 = - 4y —^ — + -qJ — ^— 



r215/l-e2\2 i75/i-e2\4 125/^ 1 -g^ \<^l Pedxc-^ 

 + |T6"V~7~; +"16 V~^"/ +"48 V""^~/ ]j a 



_„/125J_ 125 1 905 J_,905 _i25,j27.\ 

 ■*■ ^ \48 e« " 16 ^3 + 48 * e + 48 ^ 16 ^ + 48 ^') 



48 * c* + 48 e^ 48 e + 48 ^ "" 48 ^ "r 48 ^ ' 



The series equivalent to the integrals must now be substituted, in order to express 

 the quantity sought in terms containing the powers of e. 



In the first place we have these three terms, each of which is zero when the exact 

 values of Aj, A3, &c. are substituted, viz. 



^ (A. - 1 + c-») • I 



