Brown. — Phenomena of Variation. 525 



look as much like true errors as possible ; and (2) that the 

 computations of least squares are often prohibitively laborious, 

 thus practically preventing the analytical application of all 

 sorts of formulae which it may be easily possible to apply by 

 other means. 



18. The first objection will be illustrated by a coupie of 

 examples. Suppose we had a graph which consisted of the 

 curve of a phenomenon following exactly (although the com- 

 puter is not aware of this) a power-expansion formula of four 

 terms, or cubic ; and for certain reasons — say, the labour of 

 least squares — are unable to use a formula of more than three 

 terms, or parabolic. Then it can easily be seen, or proved, 

 that least squares (which becomes a problem in integration 

 in the case of a continuous curve) leads to a symmetrical 

 arrangement of the deviations the proportions of which are 

 shown in Graph A. It is pretty clear that for the observed 

 range this arrangement of deviations strikes a good average ; 

 but conceive extrapolation to be necessary, or even a terminal 

 value to be an important physical constant, would it not be 

 preferable to accept the notions which one gathers from the 

 shape of tbe curve and to extrapolate by means of some 

 such freehand curve as is drawn dotted ? The answer 

 seems obvious enough when put in this way, and yet an 

 almost precisely analogous condition of things has been the 

 cause of considerable error in a certain oft-quoted classical 

 research which the writer is recomputing by the graphic 

 process. 



19. A still more conclusive example is contained in the 

 very common case of a few datum points representing the only 

 observed facts. Here a physicist will often feel justified in 

 drawing a curve for interpolation, and will have a very strong 

 conviction of the unlikelihood of certain other curves which 

 are much different from one he might draw. If least squares 

 is followed up it is obvious that it leads to an exact represen- 

 tation of n datum points in a formula of n constants. In the 

 case of the power-expansion formula the solution is identical 

 with that of simultaneous equations. Graph B shows the 

 least-square curve passing through six points — at X = 0, 0-2, 

 04, - 6, 0'8, and 1, Y being zero at all points except 06. 

 The indeterminate question to be here answered is whether 

 there are any particular virtues about the least-square curve 

 as compared, for instance, with the clotted curve (which was 

 made by a flexible spring passing over rollers at the points). 

 Is not the interpolation here very questionable, and the ex- 

 trapolation doubtful in the utmost? It maybe here remarked 

 that the extrapolation of such formula? of high degree is al- 

 ways very doubtful, except when there is a strong conver- 

 gency. 



