[ 231 ] 



XIX. Xote oji approximations in the Theory of Probabilities. 

 By T. J. Pa. Bromwich, Sc.JD., FM.S.* 



XN the April number of this Magazine f Lord Rayleigh has 

 evaluated the formula known as Bernoulli's theorem in 

 the Theory o£ Probabilities up to terms of order 1/n 2 . 



In connexion with the revision of my book on ' Infinite 

 Series' for a second edition, I have recently noticed a formula 

 — given in (2) below— which enables us to push the approxi- 

 mations in this and similar problems as far as we please. 



The result obtained in Lord Rayleigh's problem is given 

 in (6), below : this really gives the logarithm of his formula, 

 which may be found rather easier to work with in numerical 

 calculations. 



It can be proved that when oo is large (real and positive) 

 we have the asymptotic formula % 



log{r(l + ^)}=rj 1X log^^+ilog. y + ilog(27r) 



^ 1.2x 3 . 4r 3 ^ 5 . 6^ 5 * * ' "' ' K } 



where the series is not convergent but the error committed 

 in stopping at any stage is less than the next term in the 

 series. 



Write here x = m+p, where m, p are both large, but p is 

 oE lower order than m ; in most cases it happens that the 

 order of p does not exceed >/ m. 



Then we can write 



C m+p {*m /*m+p 



log^f= logf«if+{ logfrff 



Jo Jo Jm 



= (m logm — ?») + \ log(m+t)dt. 



Jo 



In the last integral expand by the logarithmic series and 

 integrate term-by-term ; we then obtain the formula 



rm+p 



I log fdf= (m\ogm— m) 



J o 



lp 2 1 p z 1 ;/» 



* Communicated by the Author. 



t Phil. Mag. (6) vol. xxxvii. p. 321 (1919). It may be of interest to 

 •state that Lord Rayleigh had seen the results of this note just before 

 his death. 



X See, for instance, Arts. 132, 170 of the first edition of my book on 

 < Infinite Series.' The formula given there differs from the above in 

 appearance, because the integral is evaluated in the form ;v\ogx— .v. 



