()/ Variable Qtiantities. 7 



tliey have no signs of addition or subtraction prefixed. Then 

 bj tossing up a coin over and over again and calling heads + 

 and tails — , the signs + or — are assigned by chance to this 

 series of errors. About a dozen equidistant values of some 

 function (saj sine or cosine) were next taken from a Table, 

 and the errors added to or subtracted from them in order. 

 The errors may be made either small or large by multiplying 

 them by any constant. The falsified values may then be fairly 

 taken to represent a series of observations ; but we here know 

 what are the true ones. The corrections were then applied, in 

 some cases arithmetically and in others graphically, and the 

 deviations of the corrected values from the true were observed. 



In other cases a series of equidistant ordinates were taken, 

 and a sweeping free-hand curve was drawn to represent the 

 true curve, and the several ordinates of this curve were falsified 

 by the roulette and then corrected by a graphical application 

 of the rule. The general result of a good many trials was such 

 as to justify the smoothing process. Where the errors were 

 considerable the mean error was much reduced, although the 

 actual error of some ordinates was increased; w^here the 

 errors were very small the mean error was even slightly in- 

 creased. Althouo'h the dano-er of over-smoothino; was obvious, 

 and the sharpness of the features of the curve was generally 

 diminished, yet I think it was clear that the method might 

 generally be employed with advantage, especially in such 

 cases as the attempt to deduce some law from statistics or a 

 series of barometric oscillations of considerable periods. The 

 errors must be very large to justify a quadruple operation. 

 This method of trial could not be so well applied to testing the 

 case of an odd number of smoothing operations, where we are 

 left finally at intermediate ordinates. 



On the Avhole, I think the process is justifiable if applied 

 with caution. Nevertheless it undoubtedly tends to spoil the 

 results if applied to a series of points which are already in a 

 sweeping curve ; and therefore I have tried to find some other 

 process which should not have this disadvantage. This can 

 only be done by taking more than three points of the curve 

 into consideration; and therefore the process must be more 

 cumbrous. 



The method pursued was as follows : — 



Let — 2,y^; — 1, y; 0,y; l,^/^; 2, ?/2 be the coordinates 

 of five consecutive points on the curve. Suppose them to be 

 represented by a curve whose equation isi/ — a + hx + cx^ + dx^ ; 

 and make the following expression a minimum, viz. 



2(rt 4- hx + cx^ + dx^ —yf, 



