8. How many balls are there in an incomplete triangular pile of 10 layers, or strata,-each side of whose base contains 30 balls?"

Ans. 3420.

9. How many balls are there in an incomplete square pile,—the number in each side of the base being 14, and in each side of the upper stratum 6? Ans. 960.

10. How many balls are there in an incomplete rectangular pile of 12 strata,the longer side of the base containing 46, and the shorter 20 balls? Ans. 7190.

INTERPOLATION OF SERIES.

(364.) The INTERPOLATION OF A SERIES consists in finding an intermediate term in the series, which shall correspond to any proposed intermediate term in another series; the two given series having some determinate relation to each other.

For example, having given the series of Logarithms corresponding to the numbers 31, 32, 33, 34; to find, the logarithm of 33.75.

The terms of the Series in which the Interpolation is to be effected, are functions of the corresponding terms in the other series, because the values of the former depend on those of the latter; and the terms of the latter series are sometimes called the arguments of the corresponding functions.

Thus, the Logarithm of a number is a function of that number; while the number itself is the argument of the function.

Equidistant functions are such as have their corresponding arguments in arithmetical progression.

Thus, the Logarithms of the numbers 20, 25, 30, &c., are equidistant functions, since these numbers form an arithmetical progression.

In the usual applications of this subject, as in finding intermediate numbers between those which are given in Mathematical and Astronomical Tables, the series to be interpolated consists, for the most part, of equidistant functions.

When the differences of the functions are very nearly proportional to the differences of their arguments, the interpolation of a term may be effected by first differences; which is the method employed in finding the Logarithm of a number containing more than four figures (324).

When such proportion does not exist as is the case with logarithms of small numbers the interpolation may be effected by a method which regards also the higher orders of differences of the functions; and which may be called the

Differential Method of Interpolating a Series.

(365.) If x represent the term to be interpolated in a Series whose given terms are equidistant; y the number of intervals and part of an interval that the term x is removed from the first term a; and D', D", &c., the first terms, respectively, of the successive orders of differences; then will

x=a+yD'+

YY—1) D''+ y(y-1)(y-2)

2

2x3

D""+&c.

For if n be the ordinal number of the term x, we shall have

y=n-1.

Then, by substituting y for n-1 in the Formula for the nth term of a series (362), we have the Formula of Interpolation, as above.

EXAMPLE.

In latitude 40°, one degree of longitude is 45.96 miles; in latitude 41°, 45.28 miles; in latitude 42°, 44.59 miles; in latitude 43°, 43.88 miles. How many miles make one degree of longitude where the latitude is 42° 30'?

The given series of arguments is

40°, 41°, 42°, 43°;

and the given series of corresponding functions is

45.96 m., 45.28 m., 44.59 m., 43.88 m.

The term to be interpolated in the latter series, is a function of 42o 30'; and this number, in the first series, would be removed 2 vals from the first term 40°.

inter

Hence, the value of y in the Formula, in the present case, is 2. The first terms of the successive orders of differences in the series of functions, will be found to be

D'-.68; D"=—.01; D"=—.01.

Hence, by making the proper substitutions for a, y, &c., in the Formula-omitting D''', since its product would have no significant figures in the first or second decimal places-the required number is

We have thus found that, in latitude 42° 30', the length of one de

gree of longitude is 44.24 miles.

When the values of D', D", &c., do not terminate in 0, the accuracy of the interpolation will depend on the number of these terms that are used; but for most practical purposes it will be sufficient to employ only D' and D' in the computation.

This will depend, however, on the comparative value of D''', and whether this value will be sensibly increased or diminished by its multiplier in the operation.

When the term to be interpolated is one of the
equidistant Functions.

(366.) Whenever D'"', or D'""', &c., may, without important error, be taken at 0; the value of one of the intermediate equidistant Functions, may be found from the Equation, which is formed by putting the value of D''', or D'''', &c., as expressed in terms of the functions (362), equal to 0.

Suppose it were required the number of miles in a degree of longitude, in latitude 42°,. knowing that in latitude 40° a degree of longitude is 45.96 miles; in latitude 41°, 45.28 miles; and in latitude 43°, 43.88 miles.

The value of D'"', as found under proposition (362), is

d-3c+3b-a;

a, b, c, d, representing the successive terms of the given Series, which we will suppose to be the Functions

45.96 m., 45.28 m.... 43.88 m.,

in the present example, the third term c, being the one which is to be interpolated.

Taking the value of D'"' at 0, we have

The value of D'", or D'""', &c., cannot be determined without the term which it is required to find. But the series to which interpolation is usually applied, are themselves not strictly accurate; and as the successive orders of differences rapidly tend to 0, we may generally, for practical purposes, suppose D'', or D'''', to be 0.

EXERCISES

On the Interpolation of Series.

1. Find, by interpolation, the square root of 243; having given

√23=4.79583; 24-4.89898; √/25=5.

In the series of numbers 23, 24, 25, it is plain that 243 is at the distance of 13 intervals from the first term. The value of y in the Formula (365), is therefore 13.

Ans. 4.9749

2. Find, by interpolation, the cube root of 66; having given 3/64-4; 3/65=4.0207; 3/66=4,0412; 3/67=4.0615.

Ans. 4.0514.

3. Find, by interpolation, the logarithm of 103; having given the logarithm of 101=2.004321;

the logarithm of 102=2.008600;

the logarithm of 104=2.017033; (366).

Ans. 2.012837.

4. Find, by interpolation, the logarithm of 587; having given

the logarithm of 585-2.767156;

the logarithm of 5862.767898;

the logarithm of 588-2.769377.

Ans. 2.768638.

5. Find the amount of $1 at compound interest, at 6 per cent, for 12 years and 3 months; knowing that the amount for 10 years is $1.79084, for 11 years $1.89829, for 12 years $2.01219, and for 13 years $2.13293. Ans. $2. 04167.

6. Find the number of miles in one degree of longitude, in latitude 38° 15'; knowing that, in latitude 36°, a degree of longitude is 48.54 miles; in latitude 37°, 47.92 miles; in latitude 38°, 47.28 miles; and in latitude 39°, 46.63 miles. Ans. 47.12 miles.

REMARK. In the following table, in the nine right hand columns of each page, where the first or leading figures change from 9's to O's, points or dots are introduced instead of the O's, to catch the eye, and to indicate that from thence the two figures of the Logarithm to be taken from the second column, stand in the next line below.