^3 Mr. G. H. Darwin on Fallible Measiires 



and thus present small second differences. Now the proposed 

 corrections to the various observed values ai-e the quarters of 

 this series of quantities ; and thus in all probability our correc- 

 tions depend principally on the errors. The process is there- 

 fore justifiable unless the points already lie in a smooth curve. 

 The rough criterion of the applicability of the smoothing pro- 

 cess is that the second differences of the observed values 

 should not appear to conform to an}' law. 



Every double operation causes the loss of one point at the 

 beginning and one at the end ; but perhaps the best course is 

 to treat the first and last points as exact; and if the operation 

 is repeated more than twice, the second and last but one as 

 exact after one of these double operations, and so on. 



Polar Coordinates. — The preceding method is applicable 

 with equal justice to the case of polar coordinates, where the 

 ordinates are replaced by radii vectores. 



Irregular Ohservations. — With observations which are not 

 equidistant, a, strictly analogous process would be complex ; 

 but as it is empirical, a slight modification will be permissible. 

 Thus we may omit the analogue of interpolation on interme- 

 diate ordinates, and only retain the double operation. The 

 intercepts B h, C c, &c. may be bisected as before ; for this gives 

 less weight to observations which are more remote than to 

 those which are near, as it clearly ought to do. 



The corresponding numerical rule is to substitute for the 

 ordinate y^, the value 



where 



are the coordinates of the three successive points. 



The merits of an empirical rule like this must of course de- 

 pend on how it seems to work practically. I therefore devised 

 the following scheme for testing it. A circular piece of card 

 was graduated radially, so that a graduation marked x was 



720 r* 



—-7=1 e~'^^dx degrees distant from a fixed radius. The card 

 VttJo 



was made to spin round its centre close to a fixed index. It 

 was then spun a number of times, and on stopping it the num- 

 ber opposite the index was read off*. From the nature of the 

 graduation the numbers thus obtained will occur in exactly 

 the same way as errors of observation occur in practice ; but 



* It is better to stop the disk when it is spiuniug so fast that the gra- 

 duations are invisible, rather than to let it run out its course. 



