210 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS, 



we shall have 



Agn + l— 1.2.3. ..«yo ^^^^ *" * df ' 



In the first place, it may be proper to show that all the coefficients in the series for 

 Q® are positive. For this purpose integrate by parts, and the results will be 



\ ^2n+i = i.2.3...n X \'~ -Yf^^J '^^*'~ lie^S' 



Now it is evident that 



^n — 1 -^n 



d^-'' 



is divisible both by t and 1 — ^ : it is therefore zero at both the limits of the integral ; 

 so that we have simply 



Continuing to integrate in like manner, we shall find after n successive operations, 



which is obviously a positive quantity. 

 By expanding, we get 



•^" = r (1 - ^r = ^"- w. r+ ^ + w -^ • r+^ - &c. : 



and, by performing the differential operations, 



1.2. 3. ..7/ •-77r=l-/?..^4-l-T+l'** -T-) • ^ + 1 . ^ + 2 • Y72 - &c. 

 Now, because ^ = — j if we put, 



m 



1 fZ">F" 



^ W = 1 .£.3...n * "ZF' 



we shall have 



w + 1 ^ w- 1 M + 1 . n + 2 x'^ 



Another form may be given to this function ; for, without any variation in quantity, 

 t and 1 — ^ may be interchanged, not only in 



•^" = ^" (1 - ty, 



but in all its differentials, observing that the results equal in quantity will have op- 

 posite signs when the number of differentiations is odd, and the same sign when the 



'V Tit — ~ ^ 



number is even. Now if, instead of ^ = — j we substitute 1 — ^ = — - — ? we shall have 



^'W = ± |l -n-— + 



w — 1 « + 1 . w 4- 2 {m — xY 

 ^ ' ~2~ * m* 1 .2 ~ ^^- 



