176 
SHORT NOTES ON PROFESSIONAL SUBJECTS. 
112. Formulae for Computing the Number of Barrels piled in 
Magazines, by Captain S. E. Pemberton, R.A. 
Case 1.—Pyramid pile. 
$= ] + 2 + 3 . + n ; 
where 8 = number of barrels, 
n = number in bottom row. 
• O'_ n ( n + 1) 
2 . 
In an incomplete pyramid pile, 
^_ n ( n + 1) (ni — 1) in _ n 2 — m 2 + n + 
“ 2 2 “ 2 
__ (n + m) {n — m + 1) 
__ ___ ;. 
m 
where in = number in top row. 
( 1 ) 
,( 2 ) 
Case 2.—Where one end of the pile rests against an upright, and the top row 
does not touch the upright. 
S = n 2 — m 2 ; 
where n = number in bottom row, and m number in top row. 
Suppose the pile continued till m = 1. Then, 
$=2 (1 + 2 + 3 . +»)_»_ 1 
= n {n + 1) — (n + 1) = (n + 1) (n — 1) = n 2 — 1 ; .(3) 
and in an incomplete pile, 
S = (n 2 — 1) — ( in 2 —1) = n 2 — m 2 ...(4) 
Case 3.—If the top row touch the upright. 
S = n 2 — m 2 + m .(5) 
Case 4.—Where the barrels are piled between two uprights so that if bottom 
row = n, second = n — 1, third = w, and so on. 
If top row = n — 1— i.e ., if top row do not touch the uprights, 
S = mm ^ ^ (2n 1) ; .(6) 
where m = number of courses. 
When top row = n — i.e. } when top row touches uprights, 
c T f , v (2» — 1) + 1 
8 = mn — f (m — 1) = —--- } —.(7) 
z 
These results, being only calculated for piles of one barrel in depth, must of 
course be multiplied by the number of barrels in the depth of the piles. 
