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IX. On the Proof of the Law of Errors of Observations. 
By Morgan W. Crouton, F.B.S. 
Received March 24, — Read April 22, 1869. 
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 ‘ Rech ©relies sur la Probability 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 £ Theorie Analytique des Probability ’ 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 incom- 
patible 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*. 
- 2. It is remarkable that the well-known exponential function which is now pretty 
* The length of this communication may seem at variance with the statement that the proof here given is a 
simpler one than those of former writers. Still I think it will be found to he so on examination ; the length of 
the paper arises from fuller explanations being given than is usually the case. I am persuaded that the doubts 
and misconceptions which have prevailed so extensively with relation to this subject have been in great part 
occasioned by the extreme brevity and scanty explanation of the great writers who have treated it. 
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