4 Mr. G. H. Darwin on Fallible Measures 



same sign^ P will be better or worse according to the dii-ection 

 of the curvature of the curve. But if the rate of change of 

 curvature of the curve is small (as it must be assumed to be 

 to justify the smoothing process), P is very little better or 

 worse. NoWj as on the average the series of points deviate 

 as often to one side as to the other of the curve, there clearly 

 will be on the average an improvement in accuracy from the 

 substitution of P, whilst there will certainly be less abrupt 

 changes of curvature in a curve passing through P than through 

 A B. Where the points already lie on a fair curve with no 

 contrary flexure, the chance is rather more than even that there 

 will be a loss of accuracy of representation, because the sub- 

 stituted points all lie on the same side of the given ones, and 

 the only case where there is improvement is where all the 

 errors have one particular sign, and are not very small. The 

 process of smoothing must then be applied cautiously, and es- 

 pecially at maximum- and minimum-points. 



If the points P do not lie on a fair curve, the process may be 

 applied again in part of the series or along the whole line ; but 

 Avhen once our judgment leads us to think that the curve is 

 smooth enough, every succeeding operation tends to spoil the 

 representation. 



Analytically the process may be stated as follows: — 

 If ^0? yi5 3/2? <^C' be the successive given ordinates, and if <^ 

 indicates a single smoothing operation, so that (f)y^ indicates 

 the substituted ordinate corresponding to the abscissa x ; then 

 clearly </>y. = Ky.+i+yx-t), 



and generally r^^ = ( ^' +^~% ^. 



It is clear that an odd number of operations will leave us 

 Avith points on ordinates halfway between the original ones, 

 whilst an even number will leave us on the original ones. 

 There is a practical advantage in proceeding by two opera- 

 tions at a time, because it is not then necessary to draw the 

 intermediate ordinates, and because a double operation has 

 a very simple geometrical and analytical meaning. 

 From the above formula, 



<t>%=i{^j.+"''i^}- 



If in the figure MA, N B, QC are the three ordinates ^^_j, 

 ?/^, ?/j- + j, then NZ>=^(?/j,+ i+?/a;_i), and P, which bisects B^, 

 is the point to be substituted for B. 



The practical rule of construction may be stated thus: — 

 Let A, B, C, &c. be the given points ; join every point to 



