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XII. — On the Law of Frequency of Error. By Professor Tait. 



(Read 3d January 1865.) 



1. It has always appeared to me that the difficulties which present themselves 

 in investigations concerning the Frequency of Error, and the deduction of the 

 most probable result from a large number of observations by the Method of Least 

 Squares (which is an immediate consequence of the ordinary " Law of Error"), 

 are difficulties of reasoning, or logic, rather than of analysis. Hence I conceive 

 that the elaborate analytical investigations of Laplace, Poisson, and others, do 

 not in anywise present the question in its intrinsic simplicity. They seem to me 

 to be necessitated by the unnatural point of view from which their authors have 

 contemplated the question. It is, undoubtedly, a difficult one ; but this is a strong 

 reason for abstaining from the use of unnecessarily elaborate analysis, which, 

 however beautiful in itself, does harm when it masks the real nature of the 

 difficulty it is employed to overcome. I believe that, so far at least as mathe- 

 matics is concerned, the subject ought to be found extremely simple, if we only 

 approach it in a natural manner. 



2. It occurred to me lately, while I was writing an elementary article on the 

 Theory of Probabilities, that such a natural process might possibly be obtained 

 by taking as a basis one of the common problems in probabilities, viz. : — To find 

 the relative probabilities of different combinations of mutually exclusive simple events 

 in the course of a large number of trials. 



3. In fact, this is really the basis of Laplace's investigation, an elegant, but 

 very troublesome piece of analysis. With the view, apparently, of attaining the 

 utmost possible generality, he considers an error to be made up of an infinite 

 number of contributions, each from a separate source. But he assumes at start- 

 ing, that these separate contributions are as likely to be of one magnitude as 

 another, which is, to say the least, questionable ; as it seems to be inconsistent 

 with the result finally arrived at. For instance, by far the larger part of the pro- 

 bability of a given finite error is thus made to depend upon a great number of 

 infinite positive contributions, combined with a proper allowance of infinite 

 negative ones. Now, though it is not a harsh assumption to suppose that finite 

 effects should be, in certain cases, the results of additive and subtractive opera- 

 tions with infinite quantities, it does appear unlikely in the extreme, that finite 

 effects should be due to such operations in a far greater measure than to operations 

 with finite quantities. It is true that Laplace subsequently shows that the same 

 law will be arrived at by assuming any law of probability for the contributions to 



VOL. XXIV. PART I. 2 Q 



