When interpolations are made between values of a function by any of the ordinary interpolation formulae, there are generally discontinuities in the slope and curvature of the successive interpolation curves, and when, as often happens in practical work, the given values are themselves derived from rough data, these discontinuities may be considerable. To remove these breaks of continuity T. B. Sprague in 1880 devised his osculatory interpolation formula, and in the present Paper is given a brief review of subsequent contributions to the subject. From the practical point of view, the most important of the earlier contributions were those of Mr. 6. King, who successfully applied the method to the construction and graduation of national and other life tables; but while two sets of national life tables have since been constructed on the lines proposed by him, there has, for the last fifteen years, been no further development in this country either in the theory itself or in its application.
In contrast to this has been the interest in the method shown by American actuaries, particularly in recent years. Their work is briefly reviewed, and special attention is directed to that of Mr. W. A. Jenkins, who, in 1926, put forward a method of greater generality than that underlying the earlier formulae.
The weak point of the osculatory method, regarded as a smoothing agent, rests on the fact that the graduated curve is required to pass through certain predetermined points. The curve will, in fact, be constrained to take a form similar to that assumed by a flexible steel wire which is clamped at fixed points, so that, while the curve is free from discontinuities, any departure of these points from the smooth curve will be reproduced with resulting undulations. To remove this tendency to waviness, Jenkins has devised his modified osculatory method, which, while requiring the successive interpolation curves to have the same slope and curvature at their common points at the ends of each interval, does not require the curves to pass through the points corresponding to the calculated values.
His fifth difference formula has been applied to regraduate the English Life Table No. 9, Males, and in connection therewith it is pointed out that the success of the method must depend to some extent on the provision of a good set of guiding values ; for, while the curve is not required to pass through the points corresponding to these values, the closer they lie to the smooth curve the better is the graduated curve likely to be. The method recommended for adoption is one based directly on summations in fiveyear groups of the unadjusted rates of mortality. By means of a formula derived from the modified osculatory formula quinquennial values are obtained. These are treated as first approximations, and the formula is again applied to them in order to produce rates which are treated as the graduated rates. The results are satisfactory both from the point of view of smoothness and of fidelity to the data, as measured by the agreement between actual and expected deaths.
An interesting point which emerges from the various graduations and from the comparisons of actual with expected deaths is that the crude rates of mortality appear always to exceed the graduated rates for ages centering about 24, 34 . . . and to fall below them for ages centering about 29, 39 . . . ; and it is suggested that this is due to some persistent tendency to misstatement of age, and that it is also largely responsible for the waviness of graduations of population data which is generally characteristic of the application of the ordinary osculatory method.
As a further test of the possibilities of the method, and, in particular, of its smoothing powers, it has been applied to graduate the data of the Government Female Annuitant Experience 1900-1920. These data had proved very intractable to curve-fitting methods, but had been found to show an element of periodicity up to about age 90 with an apparent change in the form of the curve after that age. In connection therewith, formulae are given for applying the method to the data at the ends of the experience where the main method is inapplicable. The result is to produce a curve which is almost perfectly smooth and which closely follows the original data.
The method is essentially a graphic method in which the curves are drawn in accordance with mathematical formulae instead of being drawn by hand, and the work is arranged in such a way that it can be carried out practically from beginning to end by purely mechanical processes.