147 
744] 
TABLE OF A m 0 M 4- n (m) UP TO m = n- 20. 
it appears by inspection that, in the second column the second differences, are constant, 
in the third column the fourth differences, in the fourth column the sixth differences, 
and so on, are constant; and we thence deduce the law of the numbers in the 
successive columns: viz. this can be done up to column 7, in which we have 14 
numbers in order to find the 12th differences: but in column 8 we have only 13 
numbers, and therefore cannot find the 14th differences. The differences are given in 
the following 
Table, No. 2 (explanation infra). 
Ind. A 
1 
2 
3 
4 
5 
6 
7 
0 
1 
1 
1 
1 
1 
1 
1 
1 
2 
6 
14 
30 
62 
126 
2 
1 
12 
61 
240 
841 
2 772 
3 
10 
124 
890 
5 060 
25 410 
4 
3 
131 
1 830 
16 990 
127 953 
5 
70 
2 226 
35 216 
401 436 
6 
15 
1 600 
47 062 
836 976 
7 
630 
40 796 
1 196 532 
8 
105 
21 225 
1 182 195 
9 
10 930 
795 718 
10 
945 
349 020 
11 
90 090 
12 
10 395 
We have, by means of this Table, the general expressions of A r 0 r , A r-1 0 r , A r-2 0'', ... 
up to A }_6 0 r , viz. the formulae are 
A r 0 r -r n (r) =1, 
/r — 2\ x fr 
A , - 1 0 r 4- II (r - 1) = 1 + 2 ( ± J+l 
;T 
+ n (r - 2) = 1 + 6 ^ 1 3 )‘ +12 ( r 2 3 ) ! + 10 (’’ 3 3 ) S + 3 ( r é 3 )‘. 
&c., &c., 
where the numerical coefficients are the numbers in the successive columns of the 
?— m 
k 
is written to denote the binomial coefficient 
table; and where for shortness 
— [ . p> 01 . i^gtance, r = 10, we have 
A 8 0 10 = II (8) = 1 + 6.7 + 12.21 + 10.35 + 3.35, = 750, 
agreeing with the principal Table. It will be observed that, in the successive columns 
of the Table, the last terms are 1, 1, 1.3, 1.3.5, 1.3.5.7, 1.3.5.7.9, and 
1.3.5.7.9.11. This is itself a good verification: I further verified the last column 
by calculating from it the value of A 14 0 20 4- II (14), =6 302 524 580 as above. fihe 
Table shows that we have A , ' —)K 0’’ -r- II (r — m) given as an algebraical rational and 
integral function of r, of the degree 2m. But the terms from the top of a column, 
A0 r =l, A 2 0 1- 4-1.2 = 2 ,_1 — 1, &c., are not algebraical functions of r. 
22 October, 1879. 
19—2