THE ROYAL ARTILLERY INSTITUTION. 
163 
of x, and does not become infinite for any such or intermediate values, 
it will be found that the differences between the successive values of the 
function, and, again, the differences of these differences, &c., alter with 
a regular increase or decrease. Consequently, if the differences of any § 6. 
set of observed or incorrectly calculated numbers are found to be 
irregular, corrections must be applied to the numbers , so as to make the 
differences as regular as possible. (For examples, see Art. 55.) 
The effect upon the differences when a number is increased (or 
diminished) by unity, can be seen in the following scheme. (See 
Art. 54, and example Art. 53.) 
Let A 4 , A s , &c., be the numbers, 
Aj 1 , A 2 x , &c., a first differences, so that A ± l = A 2 — A 2 , 
Ao} = nr 3 - a 2 , 
Aj 2 , A 2 3 , &c., u second differences, so that A t 2 = A 2 — a 4 , 
&c., &c., 
and let the differences be supposed (for simplicity) all positive, so 
that each column increases downwards— i.e., A 4 > A 3 , A 7 3 > A 6 3 > A 5 3 , 
and so on. 
+ + ' -+ + 
A, 
a 2 
a 3 
V 
A, 1 
Ar 
a 2 = 
A, 3 
Ai 4 
+ 1 
A 4 
As 1 
a 3 5 + i 
A/' + 1 
Ag 4 
— 4 
a 5 + 1 
A 4 x + 1 
A 3 S-3 
V 
+ 6 
Ag 
a 5 x -i 
A 5 3 + 1 
A 4 ° + 3 
A 4 
— 4 
a 7 
Ae 1 
A^ 
V-i 
a 5 4 
+ 1 
&C., ' 
&c., 
&c. 
When A 5 is increased by 1, 
A 4 X becomes A 5 -f 1 — A 4 = A 4 ] + 1, 
V « *6-(^6+l) = A 5 l_l, 
V " (A 5 — 1) ■ — (A 4 + 1) = A 4 2 — 2, 
&c., &c. 
The alterations in each column follow the law of the co-efficients in 
the binomial theorem. 
Thus, if a column of differences requires correcting, owing to its being 
irregular, it can be seen what alterations must be made in the primary 
column. 
It may be objected that the corrected readings in Art. 55, pp. 41,42, 
do not give regular readings, since in the one case the second differences 
are 
and in the other. 
57, 57, 58, 58, 57, 58, 58, 57 ; 
24, 24, 25, 25, 25, 24, 24, 25. 
22 
