HITTING THE MAKE, ETC. 
5 
Throughout the whole of these experiments, recorded per- 
manently upon blue glazed paper, an elaborate system of test- 
ing minute working errors, by differences of a high order of 
scrutiny, approved by the first mathematicians of the age, has 
been put in practice. About 400 rounds have been tabulated : 
even corrections for the density of the air for observed read- 
ings, height of the barometer and thermometer, have been in- 
troduced. 
The precision and excellence of this mode of testing by suc- 
cessive differences may be illustrated in the following manner. 
If the velocity of the shot follows a particular law for certain 
limits, the successive differences will show its existence and any 
departure from it : just as a succession of squares and cubes 
tabulated with the slightest error can be thus detected : — 
Squares. (Error.*) 
Nos. 1 4 9 16 25 36 49 64 az 100 
1st differ. A 4 8 5 7 9 11 13 15 is is 
2nd differ. A 2 2222 22 3 0 
3rd differ. A 3 0 0 0 0 0 1 -3 
Cubes. (Error.!) 
Nos. 1 8 27 64 125 216 342 512 729 1000 
1st differ. A x 7 19 37 61 91 126 170 217 271 
2nd differ. A 3 12 18 24 30 35 aa 47 54 
3rd differ. A 3 66659 3 7 
4th differ. A 4 0 0-1 4—6 j4 - 1 
Now if the squares had been all multiplied by some constant 
coefficient, the last differences would have been equally reduced 
to zero. The same thing would have happened with the cubes. 
And, more complex still, if a new series were formed with terms 
consisting of squares and cubes, or their constant multiples, a 
series of differences would detect any deviation from a fixed 
law. 
Further, the law, being ascertained by a great number of 
experiments, slight errors of observation could be detected and 
rectified by the principles of interpolation 4 
The Bashforth chronograph, actuated by a fly-wheel of 
* An error of 1 is purposely introduced to show its exaggerated effect ; 
the third differences, 0, 0, 0, 0, 0, 0, 0, being changed into 0, 0, 1, - 3. 
t An error of 1 only is introduced purposely, viz. 342 instead of 843, and 
great departure from regularity is shown in the third and fourth differences , 
in consequence of the above error. 
% Even when a progressive law is unlmown and implied, by which a series 
of results are obtained, still by successively differencing, by means of equi- 
distant arguments , the existence of isolated errors can be speedily ‘discovered 
and corrected. 
