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425 Rh 22. Find the number of shot in an incomplete square pile of 10 courses when the number of shot in the upper course is 1005 less than in the lowest course. 23. Show that the number of shot in a square pile is one-fourth the number of shot in a triangular pile of double the number of courses.

INTERPOLATION.

530. The process of introducing between the terms of a series intermediate values conforming to the law of the series is called interpolation. An important application is in finding numbers intermediate between those given in logarithmic and other mathematical tables. For this purpose we may employ the formula used in finding the nth term by the Differential Method, giving fractional values to n.

Ex. Given log 40 = 1.6021, log 41 = 1.6128, log 42 = 1.6232, log 43 = 1.6335, $$\ldots$$ find log 40.7.

Series 1.6021, 1.6128, 1.6232, 1.6335, 1st order of differences, .0107, .0104, .0103, 2d order of differences, - .0003, - .0001, 3d order of differences, + .0002.

Substituting in formula of Art. 524, we have

log 40.7 = 1.6021 + {7}{10}(.0107)+ {7}{10}{-3}{10}{-.0003}{2}

= 1.6021 + .00749 + .000031 + .000009 = 1.6096 +.

Here log 40 is the first term (n = 1); log 41 is the second term (n = 2) ; hence in introducing the intermediate term log 40.7 we give to n a value 1.7.

EXAMPLES XLIV. C.

1. Given log 3 = 0.4771, log 4 = 0.6021, log 5 = 0.6990, log 6 = 0.7782, \ldots ; find log 4.4.

2. Given log 51 = 1.7070, log 52 = 1.7160, log 53 = 1.7243, log 54 = 1.7324, \ldots ; find log 51.9.

3. Given 5 = 2.236, 6 = 2.449, 7 = 2.645, 8 = 2.828 ; find 5.6, 7.4, and 7.74.

4. Given 3 51 = 3.7084, 3 52 = 3.7325, 3 53 = 3.7563, \ldots ; find 3 51.18.