ON QUADRATURES AND INTERPOLATION. - 367 



a very considerable portion of their range. Nevertheless its insufficiency 

 is shown by its being impossible to apply the rule to the whole range of 

 the observed life-table (including infant and senile life), without either 

 losing accuracy, or introducing a discontinuous change into the para- 

 meters. Investigations of this description, however, belong rather to 

 analysis than to mere interpolation. Their importance can hardly be 

 overrated, especially when functions involving more than one parameter 

 have to be considered ; for tables of double entry are very cumbrous, 

 and to go beyond that is practically impossible. Hence the importance 

 of Gompertz's formula, and the corresponding importance of those of 

 Jacobi's investigations * which have rendered it possible to reduce the 

 evaluation of elliptic functions, primarily depending upon three variable 

 quantities, to a combination of results obtained from interpolating double- 

 entry tables. It seemed advisable to point out the bearing of these 

 considerations upon the subject of interpolation, although their detailed 

 exposition lies outside the scope of this report. 



VI. — Interpolation and Quadeature for Uncertain Values. 



When a number of observations of a phenomenon, which can yield but 

 a single numerical value, have to be compared, the ordinary theory of the 

 errors of observation furnishes the most probable numerical amount of 

 that value, or of any given function of that value ; and this whether the 

 observations be all equally good, or have definite numerical weights 

 attached to each. A further refinement has been introduced by attaching 

 weights themselves derived from the departure of the individual observa- 

 tions from the first mean. This is a perfectly definite process, and the 

 only remark which needs to be made upon it here is, that the most 

 probable value of a given function of the result is not the same thing as 

 the given function of the most probable value of the result. 



When an unknown curve is only known by a number of points, each 

 determined subject to some unknown but appreciable error, the problem of 

 finding the curve is absolutely indeterminate, unless some assumption be 

 made as to the nature of the curve. This will be best seen by taking an 

 easy problem, in which the indeterminateness is removed by a simple 

 supposition. Let us assume that a right line has been observed, and is 

 to be plotted by means of a set of equidistant ordinates, but that upon 

 setting them off, the heads are not in a right line. It is then a perfectly 

 definite problem to find a right line such that the squares of the distances 

 of the points from it shall be a minimum, and in accordance with the 

 fundamental principle of the ordinary theory, we shall find the same right 

 line in whatever uniform direction we measure the distances. But the 

 assumption that the line through the observed points is a right line, is 

 exactly what we want to avoid in the general problem. On the other 

 hand, when points of a curve are definitely given, we may make the curve 

 determinate by assuming that its continuity is of the highest order 

 possible. In its simplest form this is effected by assuming the curve to 

 have a parabolic equation ; but this is not essential, and we may settle it 

 by circular curvature instead of by parabolic order. But whatever law 



* See Legendre, Traitc des Fonciions Elliptiqnes, vol. iii. pp. 141-2 (2nd supp. 

 art 171) ; also Jacobi, Fundamenta nova Ilieoria Functiomtm Elli^'ticarvm, pp. 139, 

 140. 



