ON THE METHOD OF LEAST SQUARES. 29 



Consequently, 



rcosuoida 1 __ &c 



J n^?f = 2 (n " p ' 



e ~4n when % is infinite. 



==_[_) 

 2 V^/ 



Therefore, 



1 _ 



2Vw7r 



-j rl 2 



and P= j= e~^ du. 



Vn-TrJo 



Now the value given for P at p. 23 is 

 P=-. = \e 



In the present case //, = 1 ; 



Ar 2 = -| /* e~*e z d = 1 ; and consequently 2yu. 2 & 2 = n. 

 Thus 



P= = e ~^ duj as before. 



V/iTT /, 



Thus Laplace's approximation coincides with the result 

 obtained by an independent method. 



This example serves to show distinctly the nature of the 

 approximation in question. 



The function p having been developed in a series of powers 

 of u, we take the principal term in the coefficient of each power 

 of w; that is, the term divided by the lowest power of n. We 



neglect for instance every such term as ^ u Zp , because we have 



U 

 a term in w 2p divided by n p . Thus we retain and neglect 



u z(p ~ 8 

 ^ , although, unless u be large, the former term is of the 



same or a lower order of magnitude than the latter. That La- 

 place's method does in a very general manner give an approxima- 



