406 Mr. M. W. Crofton on the Proof of the Law of [Apr. 22, 



chenia magna, Owen, and dwelling on the evidences of a progress from a 

 more generalized to a more specialized type of Ruminant dentition in the 

 extinct Cameloid forms succeeding each other, from the old Pliocene of 

 Nebraska to the new or Postpliocene of Mexico. 



Tables of dimensions of teeth and vertebrae of Palauchenia, Auchenia, 

 and Camelus, and drawings arranged for one folding and three 4to plates, 

 accompany the memoir. 



III. " On the Proof of the Law of Errors of Observations." 

 By M. W. Crofton, F.R.S. Received March 24, 1869. 



(Abstract.) 



The object of 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 each 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 

 occasion errors of extremely small amount as compared generally with those 

 actually produced by all the sources combined. 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 the ' Connaissance des Temps ' for 1827, and 

 also in his ' Recherches sur la Probabilite des Jugements ; ' it is also re- 

 produced in Mr. Todhunter's able ' History of the Theory of Probability.' 

 It is not therefore pretended that any new results are arrived at in the 

 present Paper. Considering, however, the importance and celebrity of the 

 question, and the refined and difficult character of Poisson's analysis, it will 

 not probably be deemed superfluous to show how the same law may be 

 demonstrated with equal generality, in a much more simple and elementary 

 manner. The difficulty of the general proof seems indeed to have been so 

 extensively felt, that several attempts have been made to simplify it. 

 However, so far as the present writer is aware, no proof has been given, 

 except Poisson's, which is not open to grave objection, as based upon 

 unjustifiable assumptions, or as unduly limiting the generality of the in- 

 vestigation. 



The mathematical reasoning in this Paper is based entirely on the above- 

 mentioned hypothesis as to the causation of error, namely, that errors 

 in rerum naturd result from the superposition of a large number of minuter 

 errors arising from a number of independent sources. The laws of these 

 elementary errors are supposed entirely unknown, no further restriction 

 whatever being imposed on the generality of the investigation ; as would be 

 the case, for instance, were we to assume (as has sometimes been done) 

 that each independent source gives positive and negative errors with equal 

 facility. To decide fully how far the above hypothesis (which seems now to be 

 generally accepted) really agrees with facts, is an extremely subtle question in 



