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VIII. — Factorials and Allied Products with their Logarithms. By Frank 
Robbins, F.R.A.S. Communicated by Professor Whittaker. 
(MS. received October 10, 1916. Read November 20, 1916. Issued separately October 20, 1917.) 
When any one of the following tables has been given in the past to an extent 
useful at the present time, it will be found by the computer that the necessary 
volume is only with difficulty accessible and hardly ever to be purchased. That 
these tables are obviously useful is perhaps all that need be said in justification 
of their more extended computation. 
The first table contains the logarithm (with eighteen decimal places) of factorial n. 
The limits of n are 1 and 120. 
If X = 2.4.6 . . . (2n — 2)(2n), then in the second table will be found log A 
also with eighteen decimal places. 
The limits of the argument (2 n) are 2 and 120. 
Similarly, if /x, = 1 .3.5 . . . (2n — l)(2n + 1), the third table gives log u for 
values of (2 n + l) from 3 to 119. 
The connection between these three functions is 
ic! = A ? x fi x _ x 
when x is even, or 
x ! = A a ._ 1 x 
when x is odd. 
These relationships were used to test the accuracy of n\, which had been 
obtained in the usual way by continual addition of logs. The agreement obtained 
step by step was complete, and it may be taken as proving both the accuracy of 
X and u as well as that of the factorial in the unabridged tables. 
For example, the last respondent from Table II, when added to the last from 
Table III, gives 
198-82539 38472 19721 542, 
in exact agreement with log 120!, found independently and given in Table I. 
These three functions were in the first instance based on logs to fifteen decimal 
places taken from Houel (l) for the integers from 2 to 100, and from Leonelli (2) 
for those between 100 and 110. As a check upon these, comparison was made with 
a table giving eighteen decimal places specially prepared by the aid of G-ray (3). 
When cut down to fifteen places complete agreement was observed between Houel 
and Leonelli on the one part and Gray on the other, and it was felt that the logs 
in hand were fit for the purpose in view, and that the only faults of which these 
three completed tables could be suspected were those unavoidably occurring in the 
end figures and due to the missing sixteenth place. Towards a forecast of this 
TRANS. ROY. SOC. EDIN., VOL. LII, PART I (NO. 8). 27 
