THE ROYAL ARTILLERY INSTITUTION. 
177 
been made by differencing, the ranges for all intermediate elevations may be 
found by interpolation. Eor much valuable information on the mathematical 
treatment of such subjects, I must refer the reader to three papers by Mr 
Woolliouse, “On Interpolation, Summation, and the Adjustment of Nume¬ 
rical Tables.” 1 He remarks, “ If in the direct calculation of a table, the 
values to be tabulated depend upon a complicated function, difficult of calcu¬ 
lation, considerable labour and corresponding risk of error may usually be 
avoided by computing the values for a certain succession of equidistant 
intervals , and afterwards determining by interpolation all other values that 
may be requisite to complete the table and adapt it to the practical purposes 
for which it is constructed. In such case the method is made use of as a 
powerful auxiliary at the option of the computer. If, however, the only 
available data are given values at certain regular intervals, and the mathemati¬ 
cal form of the function is unknown, then the process of interpolation under 
some of its modifications, becomes absolutely indispensable , and its true appli¬ 
cation is necessarily of great importance where accuracy is required,” p. 61. 
And again, “ When a series of quantities which depend upon a fixed law, 
whether known or implied , follow each other in due order of succession, the 
general accuracy of their numerical values may be satisfactorily tested by 
observing the regularity of the progression of a suitable order of differences. 
If the tabular quantities are the results of calculation from a given formula 
with equidistant arguments, by differencing them up to a certain order, the 
existence of an isolated error, if one should exist, is thus prominently 
exhibited, and therefore speedily detected,” p. 136. “If, for example, a 
number of errors, each of which may be of any magnitude, were promiscu¬ 
ously introduced into a page of a table of logarithms, or in any table com¬ 
puted mathematically according to a given formula or generic law which is 
still inherent to the general mass of values, the preceding rules, when applied 
to the differences would necessarily have the effect of practically eliminating 
the several errors and restoring the table to its original state of accuracy,” 
p. 142. 
The application of the above methods in this particular case is of a very 
simple kind, because the loss of velocity, both of the fly-wheel and of the 
shot, goes on very regularly during the short time we are concerned about 
them. The whole process is shewn applied to Round 1, and the results are 
given for all the other rounds. The manner of treating the varying velocity 
of rotation of the cylinder may be illustrated by reference to an ordinary 
table, say of logarithms. In this case there is not a uniform increase of 
logarithms corresponding to a uniform increase of numbers, and yet the 
tables are so arranged and provided with tables of differences, that there is 
not the slightest difficulty in finding the logarithm of any number, or the 
number corresponding to any logarithm within the limits of the table. 
Suppose that the following logarithms of numbers have been obtained by 
independent calculation, some of which may be erroneous. They may be 
tested by taking successive differences, as in (I ); then corrected so as to 
make the column A 2 regular, and re-differenced as in (II). 
* Tl*p Assurance Magazine, Vol. XI, p. 61-68; 301-332; and Yol. XII. p, 136-176. 
