x.u.1st Diff.2nd Diff.3rd Diff.4th Diff.
++++
65°2.14451 732 16
10153 96
66°2.24604 828 19
10981 115
67°2.35585 943 18

To find u for x = 66° 23′, we have θ = 23/60 = .3833333. The following shows the full working: in actual practice it would be abbreviated. The operations commence on the right-hand side. It will be noticed that two extra figures are retained throughout.

u0.μδu0.δ2u0.μδ3u0.δ4u0.
2.24604 +1056700+82800+10550+1900
−   1758−   158
————————————
c1 = +1054942c2 = +82642c3 = +10550c4 = +1900
P1θ = +410567½P2θ = +  161021⁄3P3θ = + 13711⁄8c4θ = +  182
————————————
uθ = 2.28710 P1 = +1071044P2 = +84013P3 = +10732

The value 2.2870967, obtained by retaining the extra figures, is correct within .7 of .00001 (§ 8), so that 2.28710 is correct within .00001 1.

In applying this method to mathematical tables, it is desirable, on account of the tabular error, that the differences taken into account in (4) should end with a difference of even order. If, e.g. we use μδ3u0 in calculating c1 and c3, we ought also to use δ4u0 for calculating c2 and c4, even though the term due to δ4u0 would be negligible if δ4u0 were known exactly.

4. Geometrical and Algebraical Interpretation.—In applying the principle of proportional parts, in such a case as that of Example 1, we in effect treat the graph of u as a straight line. We see that the extremities of a number of consecutive ordinates lie approximately in a straight line: i.e. that, if the values are correct within ±½ρ, a straight line passes through points which are within a corresponding distance of the actual extremities of the ordinates; and we assume that this is true for intermediate ordinates. Algebraically we treat u as being of the form A + Bx, where A and B are constants determined by the values of u at the extremities of the interval through which we interpolate. In using first and second differences we treat u as being of the form A + Bx + Cx2; i.e. we pass a parabola (with axis vertical) through the extremities of three consecutive ordinates, and consider that this is the graph of u, to the degree of accuracy given by the data. Similarly in using differences of a higher order we replace the graph by a curve whose equation is of the form u = A + Bx + Cx2 + Dx3 + ... The various forms that interpolation-formulae take are due to the various principles on which ordinates are selected for determining the values of A, B, C ...

B. Inverse Interpolation.

5. To find the value of x when u is given, i.e. to find the value of θ when uθ is given, we use the same formula as for direct interpolation, but proceed (if differences beyond the first are involved) by successive approximation. Taylor’s Theorem, for instance, gives

θ = (uθ − u0) ÷ (c1 + c2 θ+ ...)
2!

= (uθ − u0) ÷ P1