ON QUADEATDRES AND INTERPOLATION. 325 



intermediate values, are siraply and absolutely indeterminate. If all 

 that is known of a curve is that it passes through n equidistant points, 

 nothing is really kno^vn of the curve. If the points are in a straight 

 line, it may undulate in any manner betvreen them. Nevertheless, it will 

 often be interesting and useful to consider the simplest case, in which 

 the undulations arc minimised. Many physical problems are known to 

 contain no undulatory element, and therefore their graphical solution may 

 reasonably be assumed to be the simplest curve, cleared of undulations, 

 which will pass through the points. When these are in a straight line, 

 the solution is obvious. It is not so if they are otherwise distributed. If, 

 for instance, three points in a plane be taken generally, the simplest curve 

 through them, having regard only to its intrinsic qualities, is a circle ; 

 but there are many conceivable conditions under which a catenary, or a 

 parabola, might be more probable. The uncertainty appears to be of the 

 same order as the selection of an independent variable. In the practice 

 of experienced draughtsmen, as well as in theory, there is no rule except 

 the general avoidance of discontinuity, and there are not wanting cases in 

 which a local discontinuity is the simplest interpretation. When a base 

 line is assumed, and ordinates are measured with reference to that, it is a 

 question whether the assumption that the second differential is constant is 

 not as good as the assumption that the curvature is so. The question is 

 not very material, except in extreme cases, where the whole process 

 becomes one of bare probability rather than of approximation. In the 

 ordinary application of the calculus of finite diiierences, the work is good 

 for nothing when any extreme is approached. 



The theory of interpolation by means of finite differences appears to 

 be, historically, a mere extension of the rule of 'proportional parts,' 

 namely, that part- way between the arguments corresponds to part-way 

 between the entries — the part-icay being in each case proportional. This 

 is not exact where one increases or diminishes uniformly, and the other 

 does not ; but it was vei'y early noticed that it was approximately true for 

 most tables, and the more nearly true in proportion as the interval between 

 the successive arguments was diminished, the approximation not being in 

 the direct ratio of the diminution of interval, but nearly as the square of 

 that diminution. 



Whether few or many orders of differences be taken, the value of the 

 process depends upon the sufficiency of the convergence. If that be 

 secured, the selection of the process is a mere matter of convenience. 

 Without it the process fails. 



One of the best examples of this is to be found in the quadrature of a 

 circular segment, by ordinates set off from its chord. The semicircle, 

 however many difierences be used, always gives a bad approximation, 

 even if a great number of ordinates be taken. Regarding the question 

 analytically, it appears that the differential coefficients of the ordinate are 

 all infinite at the limits of integration, and the convergence necessarily 

 fails. Geometrically, it is an attempt to represent a curve, which is 

 parallel to its ordinate, by a curve which never can be so within finite 

 range. If, however, instead of taking a semicircle, a smaller segment be 

 taken, there is no theoretical objection to the process ; convergency is 

 secured, and practically the requisite approximation is obtained. 



The difficulty above indicated has presented itself in the case of a 

 perfectly continuous curve, with everything ascertained, so that exact 



