§ 1. Introductory. Linear interpolation between two values of a function u a and u b can be performed, as is well known, in either of two ways. If the divided difference (u h -u a )l(b -a), which is usually denoted by u (a, b) or u (b, a), is provided, or its equivalent in tables at unit interval (the ordinary difference), we should generally prefer to use the formula which is the linear case of Newton's fundamental formula for interpolation by divided differences. If differences are not given,

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... but a machine is available, then the use of proportional parts in the form of the weighted average (1.2) the linear case of Lagrange's formula, is actually more convenient, since it involves no clearing of the product dials until the final result is read. In the usual case of data tabled at unit intervals the method of (1.2) is particularly simple, since the divisor b -a is then unity or some small integer, and the division can be performed mentally. Even in the case of unequal intervals the process is convenient for machine, for the secondary dial which registers the turns, if it possesses " tens' transmission," will automatically add the multipliers b -x and x -a in (1.2), and will thus show the proper divisor b -a, which can then be used in division upon the productsum as registered by the product dials. For example, a linear interpolate M 0 . es3 can be computed from •u 0 and Mi as u o . m = 0-317 M O +0-683 M,; from M o and u s as u o . m = (2-317 M 0 + 0-683 « 3 )/3; from Mi and M 8 as u Q 683 = (2-317 M, -0-317 M 3 )/2, and so on.