390 Mr. Ivory on the Tlieory of the Astronomical Refractions. 



the quantities rejected being plainly of no account relatively 

 to those retained. Further, because co is always less than 1, 



^TZ^ — ^^ contained between a and a (1 + 2 a); and it may 



be taken equal to a, or to the mean value a (1 4-a). Thus 

 we have 



d.oO = sm 9 X — / 



\/cos^fl + 2 — + 3 — 



Again, the formula (9.) gives 



- &c. \ . 

 and if we make 



/^ 1 _ L(l+i8T^) _ . 



a . p' . u a 



S — = -^ = A, 



p t 



we shall have 



s 

 — = z ^ 

 a 



. = « - ^ (1 -C-) -/. ^^^ -/' ^4;^^ - &c. 



Let *• (m) stand for all the terms in this value of x except the 

 first, so that 



J7 = w — *• (m ) : 



from this we deduce by Lagrange's theorem, 



^-"= 0-^ - .- (*) Gr) - 1 . ^-^^f ^^^ - &c. 

 consequently, 



J « J r d.tr^^ix). 1 dd.c-^^^ix) , _ 

 duc-"=zdxc-^ + 3 —'dx + ^. , ^ c?.r+&c. 



By means of the values that have been found, the differential 

 of the refraction can be expressed in terms of one variable x. 

 In making the substitutions, the smallest term of the radical 

 quantity is to be neglected in all the terms o£ du c~", except 



