LAPLACE. -561 



their results without translating them into more usual mathema- 

 tical language. It has been remarked by R,. Leslie Ellis that, 

 " It must be admitted that there are few mathematical investiga- 

 tions less inviting than the fourth Chapter of the Theorie des 

 ProhahiliUs, which is that in which the method of least squares 

 is proved." Camhriclge Phil. Trans. Vol. viii. page 212. 



In the Connaissance des Terns for 1827 and for 1832 there 

 are two most valuable memoirs by Poisson on the probability of 

 the mean results of observations. These memoirs may be de- 

 scribed as a commentary on Laplace's fourth Chapter. It would 

 seem from some words which Poisson uses at the beginning — 

 j'ai pense que les remarques que j'ai eu I'occasion de faire en 

 1 etudiant, — that his memoirs form a kind of translation, Avhich he 

 made for his own satisfaction, of Laplace's investigations. Poisson 

 embodied a large part of his memoirs in the fourth Chapter of his 

 Recherches sur la Proh.... 



We shall begin our account of Laplace's fourth Chapter by 

 giving Poisson's solution of a very general problem, as we shall 

 then be able to render our analysis of Laplace's processes more 

 intelligible. But at the same time it must be remembered that 

 the merit is due almost entirely to Laplace ; although his pro- 

 cesses are obscure and repulsive, yet they contain all that is 

 essential in the theory : Poisson follows closely in the steps of 

 his illustrious guide, but renders the path easier and safer for 

 future travellers. 



1002. Suppose that a series of s observations is made, each 

 of which is liable to an error of unknown amount ; let these errors 

 be denoted by e^, e^, ... e^. Let E denote the sum of these errors, 

 each multiplied by an assigned constant, say 



required the probability that E will lie between assigned limits. 



Suppose that each error is susceptible of various values, posi- 

 tive or negative, and that these values are all multiples of a given 

 quantity to. These values will be assumed to lie between aw 

 and /S&), both inclusive ; here a and (3 will be positive or negative 

 integers, or zero, and we shall suppose that a is algebraically 



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