July 21, 1923] 



NA TURE 



97 



Commencing in a similar way to that of Mr. Soper, 

 we have 



log(w + i)!- log(« + ^-i)!-log(« + ^), 



where n\ is generally r(w+i). 

 Now 



log (w + -j !-log (w+ -- I j! = eWa(i-e-D) logw!, 



where D is the differential operator. 



^ ' D ^ 2 12 720 -' 



\ 3 ! a^ 4 a 12a/ 



\ 720 240^ I2a» 4 lav -* 



^log(n + j)^(n + ^)(log(n + i)-i}, 



Dlog(n + ^)=^^^^. 

 .•.logn!=(n + i){log(n + ^)-i} + (i-^)log(n+i) 



I /l , I -a\ 

 + 2(6 + ".^ ) 





''"\3!a3~4a2 12a/ f 



\ a j 



\ ^60 12a* 6a* I2av 



360 



+ a constant. 



{-!}• 



, ,— /w+(l/a)\«+W») / i\ 



(a— 2)/2a 



xexp, 



l/6 + (l-a)/a2 



ri/6+ 

 L 2 1« 



+ 



]. (.) 



It will easily be seen that this reduces to Mr. Soper's 

 form if a is taken to be equal to 2. 



As a first approximation to the value of n ! we have 



To make this the best possible first approximation, 

 I is necessary to choose a so that the first term of 

 the exponential series is zero, i.e. 



I , I -a 

 o a' 

 is an equation for determining a, i.e. 

 a^ - 6a + 6 = o. 



The roots are 3+ sj^ox 473205081 and 1-26794919, 

 Approximately these roots are 19/4 and 5/4. 



To decide which of these two values would be the 

 better, the values of the coefficients of the next two 

 terms of the exponential were determined for each 

 value of a, and it was found that these values were 



NO. 2803, VOL. 112] 



practically of the same order of magnitude. I have 

 chosen to take the lower value, because {tc+ (i/a)} will 

 be greater for that value. 



[At first it occurred to me that the desired result 

 would be obtained by making the first term involving 

 a in the exponential a minimum ; but although a 

 minimum it might be negatively large, so this criterion 

 had to be ruled out. However, it was noticed that 

 a = 2, which Mr. Soper uses, is practically the value 

 of a which makes this term a minimum, especially for 

 the larger values of n. 



The condition for a minimum is that a should 

 satisfy the equation 



a2(6w+ l) - I2aw - 6 = 0, 



i.e. a would be a function of n. 



It is the positive root which concerns us, and it 

 will be seen that as n increases this root tends to the 

 value 2, 



a = 2 + 



4« 



(61^+Ty ^PP^°^- 



Thus for the range of a values which makes the 

 first term of the exponential negative, a = 2 is the 

 worst possible choice in finding a good first approxi- 

 mation.] 



Taking a- 3 - ^/3, our series for n\ beconies 

 n != V2t( I [n + b)-" 



f _ 0-0080,1875 0-0004,6296 



(2) 



where 6 = 0-7886,7513, 6-c=J, £ = 0-28^6,7513. 



The value a = 5/4 was used in some calculations, 

 and although the series then looks simpler, there is 

 really nothing to be gained by taking this value ; 

 this is especially so for the computer who has a 

 calculating machine. It will be noticed that our 

 first approximation in (2) will be affected by an error 

 of the order of i[i2^n^ of its own value. 



First approximation : 



!= j^{-±^y\^t)-'. 



(3) 



This approximation was tested on a comparatively 

 small value of n, n=io, log 10 ! = 6-5597931, i.e. 

 10 1 = 362,9051. 



Mr. Soper's first approximation J2ir{{n + ^)le}"+'! 

 gives log 10 ! = 6-5614855, i.e. 10 ! = 364,3221. 



The correct value is 362,8800 ; the error in the 

 first case is only 251, while the error in the second is 

 14,421. 



Extending the idea, we come to consider the 

 Second Approximation. In the British Association 

 Report for 1883, p. 407, Prof. A. R. Forsyth deduces 

 a very pretty result for n ! : 



n\= sj2ir-y^ ^ '-] ... (4) 



This compact result is obtained by a process which 

 is essentially the same as the above, but applied to 

 the second term of the exponential instead of the first. 

 If we attempt to find a so that this term may be 

 zero, it is necessary to solve a quartic in ija[ = x). 



^6x* - 24^;^ - 2^x^ + 1 2;»r + I = o. 



[The term we are considering is 



L2\3a3 2aa"^6a/ 2!U\6' a^ J J J ( i\i^' 



\ a) 



C 2 



