744] 
table of A m 0” + n (m) up to m = n = 20. 
145 
of the foregoing theorem, which is used in the following form \ viz. any column of 
the table for instance the fifth, being 
A, then the following column is A, 
B, 
2 B + A, 
C, 
3 G + B, 
D, 
4 D + C, 
E, 
oE + D, 
+ E; 
and then we obtain a good verification by taking the sum of the terms in the new 
column, and comparing it with the value as calculated from the formula, 
Sum = 2A + SB + 4(7 + 5D + 6E. 
Observe that, in the two calculations, we take successive multiples such as 4D and 
5D of each term of the preceding column, and that the verification is thus a safe 
guard against any error of multiplication or addition. 
Table, No. 1, of A m 0 w -rII(m). 
< 
a 
t—1 
0 1 
0 2 
0 3 
0 4 
0 5 
0« 
0 7 
0 8 
0 9 
0 i° 
0 11 
0 12 
0 13 
0 14 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
o 
1 
3 
7 
15 
31 
63 
127 
255 
511 
1 023 
2 047 
4 095 
8 191 
3 
1 
6 
25 
90 
301 
966 
3 025 
9 330 
28 501 
86 526 
261 625 
788 970 
4 
1 
10 
65 
350 
1 701 
7 770 
34 105 
145 750 
611 501 
2 532 530 
10 391 745 
5 
1 
15 
140 
1 050 
6 951 
42 525 
246 730 
1 379 400 
7 508 501 
40 075 035 
6 
1 
21 
266 
2 646 
22 827 
179 487 
1 323 652 
9 321 312 
63 436 373 
7 
1 
28 
462 
5 880 
63 987 
627 396 
5 715 424 
49 329 280 
8 
1 
36 
750 
11 880 
159 027 
1 899 612 
20 912 320 
9 
1 
45 
1 155 
22 275 
359 502 
5 135 130 
10 
1 
55 
1 705 
39 325 
752 752 
11 
1 
66 
2 431 
66 066 
12 
1 
78 
3 367 
13 
1 
91 
14 
1 
15 
16 
17 
18 
19 
20 
1 
C. XI. 
19