164 
MINUTES OF PROCEEDINGS OF 
Such irregularities are to be met with in all calculations by -decimals, 
owing to the omission of the remaining places of decimals. Had more 
accuracy been required in either of the cases just mentioned, an addi¬ 
tional place of decimals might have been inserted by a further correction 
of the differences, as follows :— 
Bashforth, p. 41. 
Further corrected. 
74*889 
77-485 
80-138 
82-848 
85*616 
88-442 
91-325 
94-266 
97-265 
100-321 
+ A x 
2-596 
2-658 
2-710 
2-768 
2-826 
2-883 
2941 
2- 999 
3- 056 
+ Ao 
57 
57 
58 
58 
57 
58 
58 
57 
74-8892 
77-4850 
80 1380 
82-8483 
85-6161 
88-4415 
91-3246 
94-2656 
97-2645 
100*3215 
+ Ax 
2-5958 
2*6530 
2-7103 
2-7678 
2-8254 
2-8831 
2-9410 
2- 9909 
3- 0570 
+ a 2 + A 3 
572 
573 
575 
576 
577 
579 
579 
581 
1 
2 
1 
1 
2 
0 
2 
The actual w’ork of this correction need not be inserted here, as it is 
to a certain extent laborious, and the principles have been fully 
explained. 
For further explanations concerning differences that are not all posi¬ 
tive, see below (§5). 
4. Prelude to Principles oe Interpolation. 
(Chap. III., Arts. 54, 65; and Appendix III.) 
There is a common process for summing series, an explanation of 
which may be of use here, since it involves the principle of interpolating. 
Let be the sum of n layers of a triangular pile of shot; then the 
first differences are the numbers in the respective layers—namely, 
1, 3, 6, 10, 15, &c. 
2 0 
Si 
Sr 
+ Aj + A 2 + A3 
1 
3 
6 
10 
15 
2 
3 
4 
5 
1 
1 
1 
The third difference is found to be constant ; then the rule is 
+ 
n . n — 1 
1 . 2 
Ao + 
n . n — 1 . n — 2 
1.2.3 
+ 
— 1 . n — 2 . n — 3 
1 .T. 3 . 4 
A 4 + &c. 
