ON THE METHOD OF LEAST SQUARES. 33 



As for one particular law of error (that considered in p. 15), 

 the results of the method of least squares are the most probable 

 possible ; and as the function by which this law of error is ex- 

 pressed occurs in Laplace's demonstration of that method, it has 

 been thought that his approximation involved an undue assump- 

 tion, and that in fact his proof was invalid unless that particular 

 law of error was supposed to obtain. 



It is easily seen that the method of least squares can give the 

 most probable results only for that law of error (if we except 

 another which involves a discontinuous function). Mr Ivory 

 attempted to show that Laplace's conclusions might be applied 

 to prove that the results of the method were, in effect, the most 

 probable possible, and thence drew the inference which I have 

 already mentioned. After some consideration, I have decided on 

 not entering on an analysis of his reasoning, which it would be 

 difficult to make intelligible, without adding too much to the 

 length of this communication. It is set forth with a good deal 

 of confidence ; Laplace's conclusions are pronounced invalid on 

 the authority of an indirect argument, and without any examina- 

 tion of the process by which he was led to them. I may just 

 mention that in the whole of Mr Ivory's reasoning, the proba- 

 bility that 2/xe is precisely equal to any assigned magnitude, is, 

 to all appearance at least, considered a finite quantity, though it 

 is perfectly certain that it must be infinitesimal. 



It would seem as if he had taken Laplace's expression of the 

 probability in question, viz. 



without being aware that in Laplace's notation I and a are in- 

 finite, and that consequently the expression is infinitesimal. 

 (Vide TillocJis Magazine, LXV. p. 81.) 



MR IVORY'S DEMONSTRATIONS. 



They are three in number. Two appeared in the sixty-fifth, 

 and a third in the sixty-seventh volumes of TillocJis Magazine. 



The aim of all three is the same, namely, to demonstrate the 

 rule of least squares without recourse to the theory of proba- 



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