Find the number of balls required to complete

14. A square pyramid, having 144 balls in the top layer.

15. A triangular pyramid, having 5 balls in a side of the top layer. 16. A rectangular pyramid, having 22 and 8 balls respectively in the length and breadth of the top layer.

17. How many balls in an incomplete rectangular pyramid, having 16 and 12 balls respectively in the length and breadth of the top layer, and 25 in the breadth of the bottom layer?

=

18. How many layers in a square pyramid, containing 91, 7 ∙ 13, balls?

19. A rectangular pyramid of 8 layers contains 1860 balls. How many balls are in the top row?

20. The number of balls in a square pyramid, increased by 91, is equal to twice the number in a triangular pyramid of the same number of layers. How many layers in each pyramid ?

21. Show that the number of balls in any square pyramid is one-fourth of the number in a triangular pyramid having twice as many layers.

Interpolation.

20. The following example will indicate the nature of an important application of the method of finite differences.

Ex. Given 121, 224, 329, ..., find the value of (21)2. We have a,1, d= 3, d2 = 2, ds = 0, ....

3

Since the first term is the square of 1, the second term the square of 2, and so on, we may call the square of 21 the 21th term. Then,

which agrees with the result obtained by squaring.

21. Interpolation is the process of inserting between the terms of a given series other terms which conform to the law of the series.

In thus interpolating terms in a given series it is necessary, as we have seen, to give to n a fractional value in the formula for the nth term. Interpolation is extensively applied in astronomy, and is also used in computing numbers intermediate between those given in mathematical tables.

Ex. 1. From a table of square roots we obtain √42, √5=2.2361, √6 = 2.4495, √7 = 2.6458, √⁄8 = 2.8284; find √5.25.

We have

a1 = 2.0000, 2.2361, 2.4495, 2.6458, 2.8284,

d1 = .2361, .2134, .1963, .1826,

It should be kept in mind that in forming differences we always subtract a number from the number on its right, thereby sometimes obtaining negative remainders.

Since 5 is the second term and √6 is the third term, for √5.25 we take n = 2.25, = {.

Then, a2 = 2 + { (2361) + 1; }

12

(-.0227)

+1 · 1 1 (0056) + 1 · ↓ ( − 1 ) ( − 1 ) — (.0022)

13

14

In such examples it is not possible to obtain a series of dif ferences whose terms are zero. But, by taking more terms in the given series, more orders of differences, with terms nearer to zero, can be obtained. The results are approximate, and the work is to be carried only so far as it will affect the last decimal place in the values of the given terms.

EXERCISES IV.

1. Given log 30= 1.47712, log 31 = = 1.49136, log 32 = 1.50515, log 33 = 1.51851, log 34 = 1.53148; find log 31.8.

2. Given 9= 2.0801, 7/10 = 2.1544, √/11 = 2.2240, 3/12 = 2.2894, 13 = 2.3513; find 11.25.

3. The latitude of a place on the earth's surface is obtained by observing the altitude of the sun at noon, and adding to the complement of the

altitude of the sun's declination. On Oct. 31, 1896, the altitude of the sun at Philadelphia was found to be 35° 36' 37".3. The Nautical Almanac gives the following values of the sun's declination:

Find the latitude of the place of observation.

4. The length of a degree of longitude on the equator is 69.17 miles; at 10° N. latitude, 68.13 miles; at 20°, 65.03 miles; at 30°, 59.96 miles at 40°, 53.06 miles; at 50°, 44.55 miles. What is the length of a degree

of longitude at 36°15' N. latitude?

5. The altitude of a star as seen with the eye is greater than it really is, on account of the refraction of the rays of light from the star by the earth's atmosphere. The altitude of the star Fomalhaut was observed to be 19° 15' 37".3. Find its correct altitude, using the following table:

Find its weight,

6. The temperature of a litre of water is 23°.4 C. using the following table of densities, the unit being the density of water at the temperature of maximum density:

7. Nitric acid is diluted with water so that the solution contains 22.4% acid. Find the weight of a litre of the solution, using the following table of specific gravities :

CHAPTER XXXVIII.

THE EXPONENTIAL AND LOGARITHMIC SERIES.

1. In this chapter we shall give two important series, and from them derive formulæ for computing naperian and common logarithms.

The Exponential Series.

2. The expression a, in which the variable enters as an exponent, is called an Exponential Function.

3. Let

y = a2, then log, y = x log, a,

ezloga, or, a2 = ezloga.

=

(1)

(2)

wherein e is the base of the naperian system of logarithms. From (1), we obtain y If, therefore, we expand log in a series, to ascending powers of x, the result will be also the expansion of a.

4. If » > 1, the expansion of (1+1)" by the Binomial

Theorem will be a convergent series, by Ch. XXXIV., § 2, Art. 2.

however great n may be. If, then, we let n increase indefinitely,

5. It follows from the result of the preceding article that the expansion of e' can be obtained from the expansion of

n

[(1+1)*], by letting a increase indefinitely.

n

This series is found, by Ch. XXXIII., Art. 21, to be con

vergent for all finite values of x.