Miscellanea 
119 
In this case a square has to be formed, a difFerence taken, two logarithms looked out, and two 
multiplications by separate factors undertaken. It is an advantage for arithmometer work 
that our multipliers should remain the same, especially if a series of values of the function have 
to be tabled. I accordingly suggest the following formula 
log ^^^t,^^ = 0-399,0899 + 1 log "080,929 sin (iii). 
The advantages of this formula are: (i) the multiplying factors are constants, (ii) the factor ^ 
can be applied in taking out the logarithm, (iii) the recijirocals of x are tabled, whereas the 
inverse powers are not, (iv) all the terms are to be added, which saves some liability to error. 
I have slightly adjusted the constants of the sine term, so that the multiplying factors are 
fairly simple and for low values of .»■ tend to allow for the neglected — ^ term. The calculation 
requires : one use of Barlow's tables, one use of a logarithmic table, and one of a table of natural 
sines, with two applications of the arithmometer. 
As illustration take .r=7"75. 
Barlow: - = -129,0323. 
X 
Arithmometer : -129,0323 x 23°-623 = 3°-30619o=3' 18'-3717 at sight. 
Table of Natural Sines : sin 3° 18'-371 7 = -057,6719. 
Arithmometer : -080,929 x -057,6719 = -004,6673. 
/ -399,08991 
Accordingly, log = \ -444,6509 i = "848,4081. 
' ^ (-004,6673) 
Since log .»^e-^= -848,4081, we have log r (.(,■+ 1) = 4-374,7139. 
Working with 7-figure logarithms only, and using the reduction formula and Legendre's 
Tables, I find log r (.*;+ 1) =4-374,7140, which agrees absolutely within the limits of error of the 
7-figure tables. 
For this particular value of x, Formula (ii) gives log r (.<-• + 1) = 4-374,7105, not quite so good an 
agreement. 
Generally Formula (iii) is in error in the value of Y {x + \) 1 in 50,000 for x = 2, 1 in 900,000 
for x = Q, and correct to 7 figures if x=\0. It is thus more than sufficient for all statistical 
purposes, where 1 in 1000 is quite a permissible difFerence with the usual values of the probable 
errors. For the case of .37=2, the worst possible : 2 I = r (3) is given as 1-99996 instead of 2. 
Formula (i) gives 1-99948. 
We may, I think, safely conclude that Formula (iii) gives the r-function throughout the 
whole range of the variable * with an adequacy more than sufficient for statistical purposes, and 
with extremely little numerical labour. 
* Of course Legendre's Tables must still be used for x below unity. 
