144 
[744 
744. 
TABLE OF A m O n + n(m) UP TO m = n = 20. 
[From the Transactions of the Cambridge Philosophical Society, vol. xm. Part I. (1881), 
pp. 1—4. Read October 27, 1879.] 
The differences of the powers of zero, A m 0 n , present themselves in the Calculus 
of Finite Differences, and especially in the applications of Herschel’s theorem, 
/(«*)=/( l+A)^- 0 , 
for the expansion of the function of an exponential. A small Table up to A 10 0 lu is 
given in Herschel’s Examples (Camb. 1820), and is reproduced in the treatise on 
Finite Differences (1843) in the Encyclopcedia Metropolitana. But, as is known, the 
successive differences A0 W , A 2 0 n , A 3 0 n , ... are divisible by 1, 1.2, 1.2.3,... and 
generally A OT 0 n is divisible by 1.2.3 ...m, = II (to) ; these quotients are much smaller 
numbers, and it is therefore desirable to tabulate them rather than the undivided 
differences A m 0 n : moreover, it is easier to calculate them. A table of the quotients 
A m 0 n - n (to), up to m — n= 12 is in fact given by Grunert, Crelle, t. xxv. (1843), 
p. 279, but without any explanation in the heading of the meaning of the tabulated 
numbers C n k , = A n 0 k -r II (n), and without using for their determination the convenient 
formula C n k+l = nCf + C n _f given by Björling in a paper, Crelle, t. xxvm. (1844), 
p. 284. The formula in question, say 
^mQn+i ^mQn ¿pn~ IQ« 
TT(to) = m Ujmj + n (to -1) ’ 
is given in the second edition (by Moulton) of Boole’s Calculus of Finite Differences, 
(London, 1872), p. 28, under the form 
A m 0 w = to (A m-1 0 n_1 + A m 0 n_1 ). 
It occurred to me that it would be desirable to extend the table of the quotients 
A’”0 n -j- II (to), up to m = n = 20. The calculation is effected very readily by means