488 Dr. C. Y. Burton on a Physical Basis 



The method of finding the most advantageous combination 

 of measures by making the probability of the assumed value 

 a maximum was given by James Ivory * ; it appears to me 

 the most direct and satisfactory, and indeed the only method 

 which is applicable in all cases. The intimate relation which 

 has here been shown to exist between Laplace's law of error 

 and the method of least squares is also in agreement with the 

 results somewhat differently deduced in Ivory's paper. 



10. Subjective errors remain to be considered. Suppose 

 that in § 3, when the index is nearly halfway between two 

 graduations, we may not decide correctly to which graduation 

 it is nearest. Let ABC (fig. 12) represent the curve of 



Fig. 12. 



A on C 



error when the observer attempts to place this particular 

 index halfway between two graduations, O P being the 

 ordinate of no error ; so that G and A represent the 

 limits of error. Now if the distance of the index from a 

 graduation lies between +'5 centim. — OC and — *5 centim. 

 — OA, its position will be taken as certainly nearer to that 

 graduation than to any other, and the corresponding errors 

 for such positions of the index will be uniformly distributed 

 between these limits. Draw any ordinate B N of the curve 

 ABC; then the chance that in trying for the middle of a 

 division we should discard the position N for one more to the 

 right = area NBC-*- area ABC; .... (5) 



when N is to the left of A, this chance becomes a certainty. 

 The curve of error, then, for readings to the nearest centi- 

 metre will be straight and horizontal in the middle, the 

 terminal portions being found by integrating ABC from A 

 to C and from C to A. 



11. This method of treatment is not free from assumption ; 

 it would only be strictly applicable if our judgment were on 

 each occasion perfectly definite and decided, but at the same 

 time subject to a variable error following a known law of 

 frequency — like the definite but fallible indications of a 

 physical index. But it is evident that when an index is 

 sufficiently near to its true position (or to our more or less 

 biased conception of its true position), we shall have no choice 

 * Phil. Mag. 1825, vol. lxv. pp. 161 et seq. 



