IX. On the proof of the law of errors of observations

Abstract

1. So much has been published upon the Theory of Errors, that some apology seems to be required from a new writer who does not profess to have arrived at any results which were unknown to his predecessors. Nevertheless, so great, as is well known, are the difficulties of the theory, whether we seek to form a correct estimate of the principles on which it rests, or to follow the subtle mathematical analysis which has been found indispensable in reasoning upon them, that any contribution which tends to simplify the processes, without weakening their logical exactness, will probably be considered of some value. My object in this paper is to give the mathematical proof, in its most general form, of the law of single errors of observations, on the hypothesis that an error in practice arises from the joint operation of a large number of independent sources of error, each of which, did it exist alone, would produce errors of extremely small amount as compared generally with those arising from all the other sources combined. Now this proof is contained in a process given for a different object, namely, Poisson’s generalization of Laplace’s investigation of the law of the mean results of a large number of observations, to be found in his ‘Recherches sur la Probabilité des jugements,’ and which is reproduced in Mr.Todhunter’s valuable ‘History of the Theory of Probability.’ It is obvious that we should altogether restrict the generality of the proof, confining it merely to a few artificial and conventional cases, if we were to suppose each source of error to give positive and negative errors with equal facility, or to assume the law of error (even supposing it unknown) to be the same for all the sources. None of the processes, therefore, contained in the 4th chapter of the ‘Théorie Analytique des Probabilités’ are of sufficient generality for our purpose, though some writers have so employed them ; nor will the method apply here which Leslie Ellis has given in his memoir “On the Method of Least Squares” (Camb. Phil. Trans. 1844), based upon Fourier’s theorem, on account of the assumption of equal facility for positive and negative errors. The proof which follows will be found, I think, of full generality,—the only cases excluded being incompatible with the existence of the exponential law (see art. 7), and at the same time greatly simpler than Poisson’s, dispensing with his refined and difficult analysis.