Mr. J. W. L. Glaislier on a Class of Definite Integrals, 429 



x being written for \f ( ) • % integration by parts, 



it is easily shown that (58) is reduced to dependence on the 

 error-function according to the formula 



(_)«r^ 2 v-' 2 <fc: 



2' 

 1.3...(2»— 1) 



1.3... (2^-1) U 2r* + 4x d ,, ' ± 2"" 1 x 2n ~ l J* 



so that for the calculation of refraction a Table of the values of 

 Erf x was necessary. 



In 1805 the fourth volume of Laplace's Mecanique Celeste 

 was published. Chapters I. and II. of Livre X. are devoted 

 to Astronomical and Terrestrial Refraction ; and the error-func- 

 tion necessarily occupies a conspicuous place in the investi- 

 gation. On page 285 f Laplace investigates the continued 

 fraction (56). 



To give in a short space any account of the applications of the 

 function to the Theory of Probabilities would be impossible ; 

 but the results are so well known that a detailed description of 

 them would be superfluous. In the Theory of Errors of Obser- 

 vations the function is of paramount importance; and this fact 

 is the justification of the name and symbol by which it has been 

 here denoted. Laplace's great work on the subject, the Theorie 

 analytique des Probabilites, was published in 1812 ; but most of 

 the results had previously appeared in various memoirs. The 

 law of facility to which Laplace's, Poisson's, Gauss's, in fact all 

 investigations lead, is represented by e~ cx2 , so that the probability 

 of a single error of observation lying between x and x + dx is 



er 



e~ cx2 dx, and the chance of an error lying between p and q is 



(~y Ce- cx2 dx = Tr-^Erfy \Zc~Erf q */c) . 



It is unnecessary to notice the papers in which the theory of 

 errors and the method of least squares are discussed. De Mor- 

 gan's treatise in the Encyclopedia Metropolitana contains an 

 analysis of Laplace's work ; and Todhunter's ' History of Proba- 

 bilities ' supplies a commentary to it. References will be found 

 in a memoir by Ellis, Camb. Phil. Trans, vol. viii. p. 204. 



* Kramp, p. 133. 



t The reference is to the National Edition, 1845, 



