234 On approximations in the Theory of Probabilities. 



Also o.._ m _l/ x* ^ 1^_ J*L I \ 



2 (* P)-2\lUn° 12^ + 24n 4 4 >i 3 + 16 nV 



_ / ^ x4c !A 



x 30rc 5 4?i 4 + 3^ 3 / 



a 8 3.r 6 19 ^ _ 11 .r 2 1 



~288n 6 4Chi 5 + 48rc 4 24n 3+ 32n 2 ' 



which completes the verification. 



But the formula (6) above appears to be rather simpler 

 than Lord Rayleigh's for purposes of actual numerical 

 work ; it can be adapted at once to ordinary logarithms 

 by multiplying the correcting terms by log 10 e = '43429. . . . 

 Further, we can readily determine the later terms in the 

 series, if it is desired to push the approximations further in 

 any particular example. 



17th May, 1919. 



Since the above was sent to press, I have been asked by 

 Dr. EL Jeffreys to record a similar formula for the terms 

 near the maximum in the expansion of (a + b) n , where n is 

 large and a + 1 = 1. 



The maximum is readily seen to be the term nearest to 

 r = an, n — r=bn, and accordingly we wish to calculate the 

 value of 



/W-r(r+l)r(fi-r4l) L . . . (7) 

 where" r=an + %, ?i — r = bn—.v j 



Applying (2) above we now find that 



logr(r+l) + logr(n-r + l) - log T(n+ 1)- rlog a 



— {n — r) log b 



6n 2 \a 2 b' 2 ) + 2n\a b) 



1 ^/l 1\ l.r 2 /l ,1\ 1/1,1 A /0 , 

 r2n°\a" b°J 4 n~ \a~ (r / 12n\a b J ' 



* Note that x here is not the same as in Lord Rayleigh's forms : the 

 present x corresponds to \x of his work. It should also be observed that 

 an is no longer an integer. 



