Mr. Ivory on the Theory of the Astronomical Refractions. 393 

 VYi 



x(>^^(l-4 ^2 + 3^/ 3)~3/^/(>/2l-i) + ^/^) 



X -00216. 

 V 2i 

 The other parts depending on G', G", G'" are complicated ; 

 but they are troublesome more on account of the number of 

 terms they contain than from any difficulty in the integrations. 

 The following results have been obtained : 



^7^, /*^ dx dd.c-^G' J,, v^V 



' ''^ Vc vT77- • —d^^ = --^^ Til- '^ •""'' 



^ff x/ -y-x-^ ' JV^ = -/' X ^. X -02043, 



^^^ Jo ^YVx — ^:^^" = +-^^-7t;-'' '^^^^^• 



Collecting all the parts, the term sought is found, viz. 



"i±fil./*c^ dx dd.c-^^^x) 

 ^ J "" V^T^ ' dx^ 



a(l +a) Vtt 



\y 2i 



x(-00108-/'X-00142+/'2x -004.27). 



^ :/o 



To this must be added the other term, which, being inte- 

 grated in the same circumstances, gives, 



dxcr-^ ix 3 V 7r r V w^ 



7??7 ^ ¥ = - T V27 '= - Til-'' ■°°''*''- 



It thus appears that the two small terms of the expression of 

 the refraction are, together, equal to 



a (1+a) a/tt 



^^ — . (-00059-/^ X -00142 + /'^ x -00427) : 



and as «(I+«)^^^ ^ 2036"'5, 



V 2i 

 the greatest amount of both is about 1". 



The whole refraction will therefore be thus expressed : 



«=sinfl x«(l +«). /%=i£=-- . ( C-+ ^-^-^=^^^^^1, 

 ^ J \/cosH + 2ix ^ dx r 



with the assurance that the error cannot exceed 1". If we 

 substitute what * {x) stands for, we shall have 



