612 MOLESWORTH’S POCKET-BOOK
Interpolation (Sum and Differencb), &c.
A — Any term of én equidistant series of terms.
«, b, c, &c. — The first term of the 1st, 2nd, 3rd, &c,, orders
of differences.
z — The term required.
x = The distance of z from A.
x-1, x - 1 x - 2
z = A + xa + x ■■ — b + x —~ • —-— c, &c.
Example.—Find the 30th term of a series of 1, 8, 27, 64,125, &c-
1 = a
19
31
61
12 = 6
18
24
8
21
64
125
Then x being = 29 ; A = 1;
z=1+(29X7) + (29X^X12
For Interpolating a Term in a Series.
a , b, c, d, &c. — A series of equidistant terms.
h — The number of terms whose value is given. Then the
required term will be found by reducing the equation that
equation when n = 1
= 2
= 3
= 4
= 5
6 = c
6 0= Æ
a
, =7; b = 12; and c = 6,
l) + (29 X2-T X 6)=27000.
corresponds with n.
a — b — 0
a — 2b + c — 0
a — 3b + 3 c - d — 0
___x,______1 as
4(b + d) — (fl + e)
n = n
n = 4, by
a-5ö + 10c-10<Z + äe-/=0
n — 1 n— 1 n — 2 , .
a — nb + n—-— c — n—-— • —-— a, &c.,
Example.—Given a, b, d, e„ to find c; then
reducing the equation n = 4, c = ——---' - -
Given log. 2523 = 4019173 = 07 4 + = 32167148
„ log. 2525 = 4022614 = d j 1
„ log. 2522 = 4017451 =a) (a + e) = 8041784
„ log. 2526 = 4024333 = e J k J ■ —
4(b + d) - (a + e) = 24125364
Required log. 2524 = 24125364 4- 6 = 4020894.
Conversion of Rates (Multiplier = k).
x ~ = N 4- m; where M and N — the
number of times the given rates are contained in the required
rates; and m and n the times thsreguired rates are contained
in the given rates respectively.