Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 217 



expressed occurs in Laplace's demonstration of that method, it has been thought that his ap- 

 proximation involved an undue assumption, and tliat 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 shew 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, wliich 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 examination 

 of the process by which he was led to them. I may just mention that in the whole of Mr. 

 Ivory's reasoning, the probability that 2/i€ 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. 



C 4*"il'5'ral>)2 



2a\/ir V -—S 



k 



m 



(1)2 



without being aware that in Laplace's notation I and a are infinite, and that consequently the 

 expression is infinitesimal. (Vide Tilloch's Magazine, lxv. p. 81.) 



Mb. IVORY'S DEMONSTRATIONS. 



They are three in number. Two appeared in the sixty-fifth, and a third in the sixty-seventh 

 volumes of Tilloch^s Magazine. 



The aim of all three is the same, namely, to demonstrate the rule of least squares without 

 recourse to the theory of probabilities, which appeared to him to be foreign to the question. The 

 grounds of this opinion he has not clearly developed : perhaps the best refutation of it will be 

 found in the unsatisfactory character of the demonstrations which he proposed to substitute for 

 the methods of Laplace and Poisson. In common with many others, Mr. Ivory appears to have 

 looked with some distrust on the results obtained by means of this theory : a not unnatural 

 consequence of the extravagant pretensions sometimes advanced on its behalf. 



The first of his demonstrations rests upon what I cannot help considering a vague analogy. 

 In the equation of condition 



e = ax — V, 



he remarks that the influence of the error e on the value of ai increases as a decreases, and 

 versa vice: that consequently the case is precisely similar to that of a lever which is to produce 

 a given effect, as of course the length of the arm must vary inversely as the weight which it 

 supports. 



Consequently, he argues, the condition to be fulfilled, in order that the equations of condition 

 may be combined in the most advantageous manner, is the same as what would be the condition 

 of equilibrium, were a a' a" &c. weights on a lever, acting at arms e e' e" &c. This condition 

 is of course 



2ae = 0, whence 2(aa' - V) a = 0, 



the result given by the method of least squares. 



