232 D •. T. J. I'a. Bromwich on approximations 



Also 



T) 1 ?9 2 1 T? 



log (m +p) = log wi + - — - ^-s*+ o — 3 ~ • • • 



Thus on substituting in the original formula (1), we obtain 

 the result 



log {T(l + m +p) } =■ (m +p + J) log m — m+± log 27T 



■L e 2 _ L ?i 3 J_ ^ 4 _ 



+ i.2m 2Jm 2 + 3,4m 3 "' 



+ jp _1 ^ + l/_ ! 

 2m 4 m 2 6 ?» 3 



1 J??? \ m nr / 



360 m 3 \ m *"'/ 



(2) 



In the actual application it is not necessary to go beyond 

 terms of order 1/wi 3 , and regarding p as of order \/m, we find 

 from (2), the derived formula 



iog{r(i+w»-f/?)}+iog{r(i+m— p)} 



= (2m + 1 ) log m - 2m + log (2ir) +p 2 /m 

 \6 m* 2 m 1 6 in J 

 \l5 m 5 4 m 4 6 m s J 



\28 m 7 6 m 6 6 m 5 180 mV ' * W 



where the last three brackets are respectively of orders l/m y 

 ljm 2 , and 1/w 3 . 



To obtain the formula corresponding to Lord Rayleigh's, 

 we write m = ^n, p = ^v ; then, with the restrictions of the 

 problem, m+p and m—p are both integers, so that we find 



