ON THE METHOD OF LEAST SQUARES. 27 



I proceed to show that in a particular case in which the 

 value of P can be accurately determined, Laplace's approxima- 

 tion is correct. It has sometimes been thought that the intro- 

 duction of the negative exponential involves a petitio principii, 

 and is equivalent to assuming a particular law of error. It is 

 therefore desirable, and I am not aware that it has hitherto 

 been done, to verify his result in an individual case. 



Let the law of error be the same in all the observations, 

 and such that <e = ^ e+% the upper sign to be taken when e is 

 positive. 



Let ^ = p 2 = &c. = 1 , then 



(4c*i) de. ...... e+ e c?ecos a *- 



or 



coswa 



The value of p is thus given by a known definite integral, which 

 has been discussed by M. Catalan in the fifth volume of 

 Liouvilles Journal. 



It may be developed in a series of powers of u. Up to 

 w 2(n l) no odd power of u can appear in this development, for 



Q? p 



5-^- da is finite while p is less than n, and therefore the 



integral may be developed by Maclaurin's theorem. For higher 

 powers the method ceases to be applicable, and we must com- 

 plete the development by other means. But as we suppose 

 n to increase s. L the integral tends to become developable in a 

 series of even powers only of u. Thus 



r cosua , r d* 2 p a 2 , 



J.(M^^(^ 



Let 



Then 



