THE ROYAL ARTILLERY INSTITUTION. 
167 
Therefore it follows that the second differences are constant if l is 
infinitesimal. 
Let l = ds , and let dt be the consequent increment of the time— 
i.e.j that difference, A 1; which corresponds to ds; 
dt _ A x 
*'* Js~~ 1 ’ 
l being infinitesimal; 
dn d ids) d (i) d( Aj) a, 
gf' m sr = ~~i =~cr = T = coustant - 
since A 2 is constant. 
5. Principles op Interpolation. 
The principle is stated generally as follows 
Art. 64. 
, . n . n — 1 a9 . n . n — \ . n — 2 A „ , 0 ... 
*W = u* + A% + ~y ~ - . A 3 ?^ +- 2 g - • A%, + &c.; A) 
where w, is a function of x, 
u x+ i is the value acquired by w, when x is increased by I, 
'iig+ni a n h nl ; 
i.e., first by l, then by l again, and so on. 
A u x = n x+ i — w, = first difference, 
u xH —u x +- Aw,. ,.{e) 
Again, putting x + l for x in ( e ), we have 
A u„+i = — u x+ i — the next first difference; 
• • ^*+2 1 ~~ H'u-Yl ~h Aw A '4-? 
— u x -\- Aw, + A (u x + Aw,) 
= w» + 2 A u x + A 3 w,. 
(/) 
