■214 Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 



Thus 



/ 5— = /^w n - - . ?4^ + u* - &c. > 



J„ (1 + a^)" I 2 271-3 2.S.i 271 - 5 .271 - 3 j 



The coefficient of ii^'' is 



1.3...2P-1 1 



±/« 



2.3...2p 2w - 1 -2p...2re - 3 



] 1 1 



1.2. ..p. 2f 2n- 1 - 2p...2n~ 3 

 Let n become infinite, this becomes 



fin). ' ' 



1 .2...p ^iTiy' 



and we have only to determine what /(«) then becomes. 

 Now by AVallis's theorem 



/2y^l.3...(2n-i)^^^^^ 



WJ 2.4...2W-2 



1 / 2w \J 

 Therefore /"n = - when w is infinite, 



„ 1 /7r\J 

 or, /w = - - . 



i„ (1 + ay " 2 [n) I " 4^ "•" 2 " (4n)= " ""' 1 

 2 W/ 



Consequently, 



" when w is infinite. 



Therefore, 



1 -"- 



and P= I e'*" du. 



Jo 



Now the value given for P at p. 210 is 



P = 



In the present case /u = I . 



/f' = 2 Jo ""'e'rfe = 1 : and consequently 2//A--=w. 



Thus 



1 ^( £. 



J' = , / *" du, as before. 



Thus Laplace's approximation coincides with the result obtained by an independent method. 

 This example serves to shew 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 u ; that is the term divided by the lowest power of ». We 



