Discordant Observations. 365 



methods may be reduced by the following reflection. Different 

 methods are adapted to different hypotheses about the cause of 

 a discordant observation ; and different hypotheses are true, or 

 appropriate, according as the subject-matter, or the degree of 

 accuracy required, is different. 



To fix the ideas, I shall specify three hypotheses : not pre- 

 tending to be exhaustive, and leaving it to the practical reader 

 to estimate the a priori probability of each hypothesis. 



(a) According to the first hypothesis there are only two 

 species of erroneous observations — errors of observation proper, 

 and mistakes. The frequency of the former is approximately 



represented by the curve y= —^e~ h2 * 2 ; where the constant h 



v "" 

 is the same for all the observations. But the mathematical 

 law* only holds for a certain range of error. Beyond certain 

 limits we may be certain that an error of the first category 

 does not occur. On the other hand, errors of the second 

 category do not occur within those limits. The smallest 

 mistake is greater than the largest error of observation proper. 

 The following example is a type of this hypothesis. Suppose 

 we have a group of numbers, formed each by the addition of 

 ten digits taken at random from Mathematical Tables. And 

 suppose that the only possible mistake is the addition or sub- 

 traction of 100 from any one of these sums. Here the errors 

 proper approximately conform to a probability curve (whosef 

 modulus is Vl65), and the mistakes J are quite distinct from 

 the errors proper. 



Here are seven such numbers : each of the first six was 

 formed by the addition of ten random digits, and the seventh 

 by prefixing a one to a number similarly formed — 

 45, 23, 31, 50, 42, 45, 136. 



* This follows from the supposition that an error of observation is the 

 joint result of a considerable, but finite, number of small sources of error. 

 The law of facility is in such a case what Mr. Galton calls a Binomial, or 

 rather a Multinomial. (See his paper in Phil. Mag. Jan. 1875, and the 

 remarks of the present writer in Camb. Phil. Trans. 1886, p. 145, and 

 Phil. Mag. April 1886.) 



t I may remind the reader that I follow Laplace in taking as the 

 constant or parameter of probability-curves the reciprocal of the coefficient 



of x: that is =-> according to the notation used above. It is v2 times 



the " Mean Error " in the sense in which that term is used by the 

 Germans, beginning with Gauss, and many recent English writers 

 (e. g. Chauvenet) ; and it is n/n times the Mean Error in the (surely more 

 natural) sense in which Airy, after Laplace, employs the term Mean Error 

 (Chauvenet's Mean of the Errors). 



X In physical observations the limit of errors proper must, I suppose, 

 be more empirical than in this artificial example. 



