FACTORIALS AND ALLIED PRODUCTS WITH THEIR LOGARITHMS. 
169 
These elements were re-groupecl so as to produce factors of easy application — 
50 ! = (23 . 29 . 31 . 37 . 41 . 43 . 47 ) x 11 (7 3 . II 3 . 13 3 ) 
(7 2 . 17 2) ( 7 3 . 192) ( 2 9 . 3 5 ) 2 (2® . 3 13 ) 2 11 . 23 . 10 12 . 1/3 
= 63392725189 (11033033011) (14161) (123823) (124416) 2 (102036672) (2048) 23 . 10 12 . 1/3, 
and the multiplications being done, the extended value of this factorial exactly 
reproduced the result obtained step by step and so confirmed the whole table — - 
except for compensating errors. 
Comparison was made with Zimmermann (4), who stops at 20! . 
The last table — the reciprocals to eighty-five decimal places of the factorials — gave 
me most trouble. Each result, as soon as obtained by division, was checked either by 
multiplication or by repetition on the arithmometer. Apart from this, the reciprocal 
of 14! was again produced as the quotient of one by 1001 x3 s x4 s x7x8x9x 10 2 , and 
the reciprocal of 20 ! was found therefrom by a further division by 2 5 x 3 3 x 17 x 19 x 10 2 . 
The results agreed to the eighty-seventh decimal. 
Other tests applied were such as 25! x 1/25 ! , which would give unity if we could 
find the reciprocal completely. Assuming my value of 1/25 ! to be too large by 5 
in the eighty-seventh place, then multiplication by 25! would give 1 + 775 x 10~ 64 . 
By actual multiplication I obtained 1 + 2 x 10" 62 . 
So too by actual multiplication of my values for 40! and 1/40! I obtained 
1 + 36 x 1CT 40 , which indicates an error of no more than 5 x 10' 87 in my value of 
1/40! . But I relied most of all on the value of (e— l) given by the casting of these 
reciprocals. By proceeding to 1/64! and including eighty-six decimal places I found 
e = 2-7182818284590452353602874713526624977572470936999595 
749669676277240766303535475945713, 
which is too small by a unit in the eighty-fifth place, as I find by comparison with the 
value to 137 places given by Dr Glaisher (5). Lastly, these reciprocals were read with 
the table to twenty-eight significant places given in the same paper. To save space in 
the printed page, (m) indicates m cyphers between the decimal point and the first 
significant figure. 
The second and third tables are extensions of some' made many years ago to 
evaluate Wallis’s formula, 
7t 2. 2. 4. 4. 6. 6. 8... 
2 “1.3. 3. 5. 5. 7. 7 . . .’ 
a formula of engaging simplicity but of a tediousness quite wearisome if more than a 
moderate number of decimal places are required. But these tables help here ; they 
also give the coefficients in Euler’s formula for tan _1 £ and those which occur in Fi* 
and Ei*. 
Factorial n and its reciprocal have been included at the suggestion of Professor 
E. T. Whittaker for the benefit of those who, in yearly increasing numbers, 
realise that an arithmometer — in suitable work — is more speedy than a table of 
