Mr. Ivory on the Theory of the Astronomical Refractions. 499 



Investigatioji of the integral Qq. 

 We have 



*n^eda:c~^ /^^ edxc~'' 



assume, 



then, 



= \/(l- 



A = A /n_.,2\9 , A«2 ^ 



e^fj^i^e^,:^^ l^e^ + 2eH; 



m 



e dx 



~ A - = e.mdz, 



By Lagrange's theorem, 



^ = t -t% 



, , ^4 r/.*^ e^ dd.^ , g„. 

 , = / + .^ *+-.~^ +___.__ + &c.; 



consequently, 



/med.vc-'' /»! ,, ,r «<?* e^ ^*-2 , o 1 



Wherefore, if we assume 



Qo = Ai e + A3 ^3 + Ag e^ + &c., 

 we shall have 



In the first place, it may be proper to show that all the co- 

 efficients in the series for Q^ are positive. For this purpose 

 integrate by parts, and the results will be 



Now it is evident that 



is divisible both by t and 1— ^ : it is therefore zero at both 

 the limits of the integral ; so that we have simply 



A.2n4.i = — 7: • / mdtc^' . — ^ 7-. 



'^"^^ 1.2.3...W J o rfif"-' 



2K2 



A^2n+i— -,-77-5 r mdtc-*"* 



^"+' 1.2.3. ..n^ 



