THE EOYAL ARTILLERY INSTITUTION. 
165 
But 
and 
A 4 and all further differences = 0, 
Ai — 1, A 2 — 2, A 3 — 1 
“ n -f* w . n — 1 + 
n . n — 1 . n — 2 
6 
= $ + — — - + 
6 2^3 
n_~n.n-\-\.n-\-2 
“ ~~ 1.2.3 
Similarly, for a square pile, it will be found that A 2 = 1 , A 2 = 3, 
A 3 = 2, A 4 , &c., = 0, since the third difference is constant. 
, Sn . n — 1 n . n — 1 . n — 2 
••• S .V = n +-2- + -3- 
_n.n + l-, 2n+l 
” 1.2.3 * 
which is the ordinary formula. 
This process can be adopted for summing any series whatever, if any 
column of differences is found to be constant, and consequently all suc¬ 
ceeding differences vanish. 
Take one more case. 
1.3. 5+2.4.6 + 3.5. 7 + &c. 
= 15 + 48 + 105 + 192 + 315 + 480 + &c.A 2 
33 57 87 123 165. A 2 
24 30 36 42 ..... A 3 
6 6 6 ...A 4 
.’. A 1 = 15, A 2 = 38, A 3 = 24, A 4 = 6, A 5 &c. = 0; 
• 2 = 0 + 15n + ‘ • n ~~l • n ~~% _j_ 6^ . n — 1 , n—2 . n —3 
This, when reduced, will be of the form 
An + Bn% + Cn s + Bn^ y 
the fourth difference having been constant. 
The number of shot in the piles before mentioned was of the form 
An + Bri* + Cn s } 
the third difference having been constant in each case. 
It may, then, be inferred (or proved by induction) that whenever the 
r th column of differences is constant, the sum of the series— i.e., the 
