THE ROYAL ARTILLERY INSTITUTION. 
169 
positive— i.e., the original function and all the successive differences 
have been supposed to increase when “ x ” (the independent variable) 
increases; but this will seldom be found to be the case. 
The differences are all positive in the case of the series in § 4, but not 
in the example from logarithms given in Art. 53. (See also § 6.) 
All the preceding results are, however, equally true when any of the 
differences are negative, provided the proper signs are affixed. 
To avoid error, it must be remembered that negative quantities are 
increased when made numerically less, and diminished when made 
numerically greater; that is, 
— 4 > — 5 v—4=—5 + 1 
— 4 < -3 v — 4 = — 3 — 1 
Hence, in any column of positive differences (see scheme Art. 54, or 
§ 3), if they are found to diminish as they proceed— i.e. } as x increases 
—the next column will consist of negative differences. If these, again, 
are found to diminish numerically as they proceed, they are really 
increasing; therefore, the differences in the next column are positive 
and not negative. Of course, also, if negative differences are increasing 
numerically, the next column is positive. This should be carefully 
thought over, as it is necessary in order to obtain a clear comprehension 
of the nature of differences. 
Suppose that, in the scheme § 3, the numbers “ N }> in the first 
column are increasing, then the first differences “ A 39 are positive ; for 
A 1 = IV 3 — N v which is positive, since Aj > AJ. 
Suppose that the first differences are also increasing, 
•. A 2 > Ap A 1 3 = A 2 — Aj is positive; 
that is, the second differences are positive. Suppose these to be 
diminishing, 
A 2 3 < A x 3 . A x 3 = A 2 3 — A x 3 isjiegative; 
that is, the third differences are negative. Suppose, again, that these 
also are diminishing, 
••• V < V- V=(-a 2 8 )-(- V) 
= A x 3 — A 3 3 is not negative but positive. 
If these, again, are diminishing, A 5 will be negative. 
We see, then, that if all the columns of differences are diminishing, 
they will be alternately negative and positive; whereas, if all are 
increasing, all will be positive. 
It must be remarked with regard to the corrections to be applied in § 
differencing, that if any column is negative, those quantities must be 
numerically subtracted which would have had to be added if it had been 
positive; and vice versa t 
