PILING OF SHOT, AND SHELL.

Shot, and shells, are usually piled in horizontal courses, the base being either an equilateral triangle, a square, or a rectangle. The triangular, and square piles terminate each in a single ball, but the rectangular pile finishes in a row of balls.

To find the number of balls in a complete pile.

Rule.—Add the three parallel edges together; then one-third of the product of that sum, and of the number of balls in the triangular face, will be the number sought.

Note 1.—The parallel edges in a rectangular pile are the two rows in length at the base, and the upper ridge. In the square pile the same, except that the upper row is only a single ball. In the triangular pile, one side of the base, the single ball at top, and that at the back, are considered the parallel edges.

Note 2.—The number of balls in the triangular face is found by multiplying half the number in the breadth at the base, by the number in the breadth at the base plus 1.

Note 3.—In all piles the breadth of the bottom is equal to the number of courses. In the oblong pile, the top row is one more than the difference between the length, and breadth of the bottom.

Example.—To find the shot in a triangular pile, the bottom row consisting of 12 shot.

Parallel	{	12
edges.	{	1		12 ÷ 2	= 6
	{	1		12 + 1	= 13
			Triangular face		78
	3 )	14			4⅔
		4⅔			312
					52
			Answer		364

Example.—To find the shot in a square pile, the bottom row consisting of 12 shot.

	12
	1		12 ÷ 2	= 6
	1		12 + 1	= 13
				78
3 )	25			8⅓
	8⅓			624
				26
		Answer		650

Example.—To find the shot in an oblong pile, whose base consists of 18 shot in length, and 12 in breadth.

	18		18 - 12	= 6
	18			1
	7			7
3 )	43
	14⅓		12 ÷ 2	= 6
			12 + 1	= 13
				78
				14⅓
				312
				78
				26
		Answer		1118

Triangular pile.

Rule.—Multiply the base by the base plus 1, this product by the base plus 2, and divide by 6.

Square pile.

Rule.—Multiply the bottom row by the bottom row plus 1, and this product by twice the bottom row plus 1, and divide by 6.

Rectangular, or oblong pile.

Rule.—Multiply the breadth of the base by itself plus 1; and this product by three times the length of the base plus 1, minus the breadth of the base, and divide by 6.

In the following formulæ let the letter (L) denote the number in the bottom row, or the length; and (B) the breadth of the lowest course.

Triangular pile	L × (L + 1) × (L + 2)
	6
Square pile	L × (L + 1) × (2L + 1)
	6
Oblong pile	B × (B + 1) × (3L + 1 - B)
	6

The number of shot in any pile,

(whose base does not exceed 21) may readily be ascertained by referring to the following Table, [page 284].

For the square pile.—Look for the number of shot in the base, in the first vertical column on the left hand, and also in the diagonal column; and at their angle of meeting will be found the content required.

Thus 20 base gives 2870.

For the triangular pile.—Look for the number in the base row in the diagonal column, and opposite to it will be found the content.

Thus 18 base gives 1140.

For the oblong pile.—Look for the number in the length of the base in the vertical column, and the breadth of the base in the diagonal column, and at their angle of meeting will be found the content required.

Thus 17 length, and 12 breadth, gives 1040.

To find the number of balls in an incomplete pile.

Compute the number in the pile considered as complete; also the number in the upper pile, or part wanting; and the difference between the two piles thus found will be the number in the frustrum, or incomplete pile.

Table for computing the Content of any Pile, whose base row does not exceed 21 balls.
1	2	4
2	5	3	10
3	8	14	4	20
4	11	20	30	5	35
5	14	26	40	55	6	56
6	17	32	50	70	91	7	84
7	20	38	60	85	112	140	8	120
8	23	44	70	100	133	168	204	9	165
9	26	50	80	115	154	196	240	285	10	220
10	29	56	90	130	175	224	276	330	385	11	286
11	32	62	100	145	196	252	312	375	440	506	12	364
12	35	68	110	160	217	280	348	420	495	572	650	13	455
13	38	74	120	175	238	308	384	465	550	638	728	819	14	560
14	41	80	130	190	259	336	420	510	605	704	806	910	1015	15	680
15	44	86	140	205	280	364	456	555	660	770	884	1001	1120	1240	16	816
16	47	92	150	220	301	392	492	600	715	836	962	1092	1225	1360	1496	17	969
17	50	98	160	235	322	420	528	645	770	902	1040	1183	1330	1480	1632	1785	18	1140
18	53	104	170	250	343	448	564	690	825	968	1118	1274	1435	1600	1768	1938	2109	19	1330
19	56	110	180	265	364	476	600	735	880	1034	1196	1365	1540	1720	1904	2091	2280	2470	20	1540
20	59	116	190	280	385	504	636	780	935	1100	1274	1456	1645	1840	2040	2244	2451	2660	2870	21	1771
21	62	122	200	295	406	532	672	825	990	1166	1352	1547	1750	1960	2176	2397	2622	2850	3080	3311	22	2024