but one ball in each. The number
of balls in a triangular face is x(x+1)/2; x being the number in the
bottom row. The sum of the three parallel edges in a triangular pile
is x+2; in a square pile, 2x+1; in an oblong pile, 3X + 2x-2; X being
the length of the top row, and x the width of the bottom tier; or
3m-x+1; m being the length, x the width of the bottom tier.

If a pile consists of two piles joined at a right angle, calculate the
contents of one as a common oblong pile, and of the other as a pile of
which the three parallel edges are equal.

96. TABLE GIVING THE NUMBER OF BALLS IN A TRIANGULAR PILE, THE BASE OF
WHICH IS X.

VALUE OF | VALUE OF | VALUE OF | VALUE OF | VALUE OF |
-----+------+-----+------+-----+------+-----+------+-----+------+
X. | S. | X. | S. | X. | S. | X. | S. | X. | S. |
-----+------+-----+------+-----+------+-----+------+-----+------+
1 | 1 | 6 | 56 | 11 | 286 | 16 | 816 | 21 | 1771 |
2 | 4 | 7 | 84 | 12 | 364 | 17 | 969 | 22 | 2024 |
3 | 10 | 8 | 120 | 13 | 455 | 18 | 1140 | 23 | 2300 |
4 | 20 | 9 | 165 | 14 | 560 | 19 | 1330 | 24 | 2600 |
5 | 35 | 10 | 220 | 15 | 680 | 20 | 1540 | 25 | 2925 |
-----+------+-----+------+-----+------+-----+------+-----+------+

97. TABLE GIVING THE NUMBER OF BALLS CONTAINED IN A SQUARE PILE, THE
BASE OF WHICH IS X, AND IN A RECTANGULAR PILE, THE SIDES OF WHICH ARE
X AND X + N.

VALUE | DIFFERENCES | VALUE OF N.
OF +-------------+-----------------------------------
X. | 2d. | 1st. | 0. | 1. | 2. | 3. | 4. |
------+------+------+------+------+------+------+------+
2 | 7 | 9 | 5 | 8 | 11 | 14 | 17 |
3 | 9 | 16 | 14 | 20 | 26 | 32 | 38 |
4 | 11 | 25 | 30 | 40 | 50 | 60 | 70 |
5 | 13 | 36 | 53 | 30 | 85 | 100 | 115 |
------+------+------+------+------+------+------+------+
6 | 15 | 49 | 91 | 112 | 133 | 154 | 175 |
7 | 17 | 64 | 140 | 168 | 196 | 224 | 552 |
8 | 19 | 81 | 204 | 240 | 276 | 312 | 348 |
9 | 21 | 100 | 285 | 330 | 375 | 420 | 465 |
10 | 23 | 121 | 385 | 440 | 495 | 550 | 605 |
------+------+------+------+------+------+------+------+
11 | 23 | 144 | 506 | 572 | 638 | 704 | 770 |
12 | 27 | 159 | 650 | 728 | 805 | 884 | 962 |
13 | 29 | 196 | 819 | 910 | 1001 | 1092