168 
MINUTES OF PROCEEDINGS OF 
Similarly, 
w* +3Z = u x + 3 Aw* + 3 A 2 w* + A 3 w*, 
and generally, 
tlx+nl = U x + wAw* + n - - A 2 W* + &C. 
Thus equation (A) is proved, and this is the rule given in § 4 for 
summing series. 
In proving (f) the process is not one of multiplying out, when we 
say that A (w* + Aw*) = Aw* + A 2 w* ; but it is this—namely, the difference 
of (w* + Aw*) is the difference of w* + the difference of A u XJ which latter 
is (by definition) the second difference of w*. 
A (w, + Aw*) = Aw* + A 2 w*. 
Much depends upon equation (A); so that even those who are not 
acquainted with the calculus should endeavour to understand it. 
This equation being a general equation, it may excite surprise that 
the letter “ l,” to which various values may be given, occurs on one side 
only of the equation. Any difficulty on this account is easily explained 
•away by the consideration that the magnitude of Aw*_, &c., depends upon 
the magnitude of “ l,” the increment of x; therefore “ l ” is involved 
in Aw*, &c. 
Equation (A) may be supposed to be represented by a continuous 
curve, such that, for any value of “n 9> —the abscissa—the corresponding 
ordinate should represent the value of “ u x +ni Aw*, &c., being treated^* 
as constants, since they are independent of the value of “ n.’ } From 
this it can be seen that, although “ n ” is essentially an integer in the 
proof of equation ( A ), yet the error will be infinitesimal if it is con¬ 
sidered to hold for all intermediate values of “ n.” It does hold for 
“n = o”; hence it may be used for all fractional as well as integral 
values of “ n.” 
It is used with fractional values for interpolating. (See Appendix III., 
where it is numbered (I.), and transformed into other, at times, more 
convenient forms (II.), (III.), (IV.), which the reader need not trouble 
himself'to get up, unless he is studying the subject of “ finite 
differences,” since they are not essential to the due comprehension of 
Mr. BashfortlTs work). 
Equation ( A ), then, is the formula for interpolation; for if we know 
u x , u x +i, w*+ 2 i, &c., we can, by giving fractional values to “ n ,” obtain the 
values of functions intermediate to w* and w*+z. 
Examples from logarithm tables will be given on the application of 
equation (A) to interpolation, after considering what algebraical signs 
must be prefixed to the differences in all cases. 
§ 3. In the preceding investigations all the differences have been supposed 
