166 
MINUTES OF PROCEEDINGS OF 
function of n which we are considering—is a function of the r th order, 
namely, of the form 
An + Bn 2 -f- . + Rn r . 
Art. 66. Professor Bashforth found that the second differences of the times 
occupied by a projectile in passing through successive* equal intervals 
were constant. Hence, putting “ s ” for any number of intervals (cor¬ 
responding to <c n,” any number of terms), and t for the whole time of 
passing through “ s ” — i.e., for the sum of the times occupied by each 
interval (corresponding to above)—we have the very important 
expression 
t = as + 5s 2 ...( d) 
By giving “s” the values 1, 2, 3, &c., successively, the following 
values are obtained for t :— 
a -f* 5, %a -f- 4 5, 3 a -J- 95, 4t(t, -f- 165 
a + 35 a + 55 a + 75 . A 1 
25 25 . A 3 
Thus the second difference is constant, which proves the correctness 
of the formula (d ). 
Another proof can be given by the calculus as follows :— 
Since the second difference of the times is constant, it follows that 
dH 
ds 2 
= constant = 25, suppose. 
Hence , integrating, 
and integrating again, 
But t = o when s — o } 
~ = 25s + constant 
ds 
= a + 25s, suppose; 
i = as + 5s 2 + constant. 
.*. t = as + 5s 2 . 
■W 
Should it not be quite clear that the equation “ ^ = C” is obtained 
from the fact that the second difference is constant, the following 
analysis may serve to throw light upon it:— 
If for intervals l the second differences are constant, it will be seen 
without difficulty that they are also constant for any subdivisions of l .* 
* Let l be subdivided into n equal parts, 
Then, referring to the scheme in § 3, we have n differences “ 5 ” corresponding to each A 1 , which 
is equal to the sum of such n differences. 
Thus from JV 2 to JVo there are n differences, varying from ^±1 to , which differ by ; 
n n n 
therefore, as there are n of them, their differences— i.e., the new second differences—approximate to 
t. 2 
_i-, and are, a fortiori, constant if the original second differences are constant— i.e., constant to as 
u'-* 
many places of decimals as appreciably affect the result. 
