206 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



To this must be added the other term, which, being integrated in the same circum- 

 stances, gives, 



3 r dxc~^ ix 3 V T 's/~i (\c\i\Ac\ 



~ "2 '^0 viT^ >^ T - ■" ¥ * 'VTi "VTi ^ '^^^^^' 



It thus appears that the two small terms of the expression of the refraction are, 

 t^ogether, equal to 



^^^+g./^ . (-00059 -/' X -00142 +/2 x -00427) : 

 and as __ 



'('+/t^/' = 2036"-5. 



the greatest amount of both is about 1". 



The whole refraction will therefore be thus expressed : 



TV. • . /, . X /* d^ ( . d . c-"" "ir ix)\ 



V ' ^ ./ -/ cos^ ^ ^ lix \ dx r 



with the assurance that the error cannot exceed 1". If we substitute what Y {x) 

 stands for, we shall have 



</.^^ = sin^ X a(l -fa) X / / . / ^ ■ X 

 ^ ^ ' J V cos* d + 2 » X 



This expression being regular, it may be continued to any number of terms, and it 

 has the advantage of being linear with respect to the coefficients. Adverting to what 

 X stands for, it will appear that L x ^ is nearly equal to *, or to 2, that is, to the ele- 

 vation in the atmosphere ; so that, if we suppose the greatest height of the atmosphere 

 is 10 X L, or about fifty miles, the greatest value of x will be 10 ; and all the inte- 

 grals in the foregoing expression must be taken between the limits zero and 10. But 

 the quantity c~^ is so small when x has increased to 8 or 10, that the results are not 

 sensibly different whether the integrals be extended to those limits or be continued to 

 infinity. By substituting the values of the functions, the expression of ^ ^ will take 

 this form : 



^ ^ = sm ^ X a (1 + a) X < / . ,^ ^=^ + >^ / — ) o, • — 



_/ A=^==:. . (4 c-^^ -- 3 6-^ -I- .T c-') 

 +/' /. f/ ^^ . (s c- ^^ - 8 c- ^ + 7 ^ c- - - 2 0.2 c- ^ +^-) 



. •' »y /v/ cos* ^ -\- ^i X \ " 



"T" 6 ^^ ■"24'^^ + 120 /• 



