22 ON THE METHOD OF LEAST SQUARES. 



The next step is to find an approximate value of this 

 expression. 



/ +*> /-+ 30 



When = I <e cos a//,ecZe == I ^e^e = 1, 



J -ao ^ -oo 



as the error e must have some value lying between + oc . 



It is clear this is the greatest value the integral in question 

 can have, and therefore as n increases sine limite, the continued 

 product 



/ + / +00 



^> 1 6 1 cos /^ae, Jej . . . / < n e n cos f* n ae n de n 



00 -00 



decreases sine limite, (being the product of n factors each less 

 than unity) except for values of a differing infinitesimally from 

 zero. 



eVe, K* = f 

 ^o 



and develope each of the cosines in the above-written continued 

 product. It is thus seen to be equal to 



- a 



- &c. 



Again, n being very large and ultimately infinite, it is 

 evident that 2//,V is of the same order of magnitude as n, while 

 ^/^a&i^a is of the order of w 2 , the former term of the coefficient of 

 a 4 may therefore be neglected in comparison with the latter, 

 which again may be replaced by \ (2^ 2 & 2 ) 2 , from which it differs 

 by a quantity of the order of n. Similar remarks apply with 

 respect to the higher powers of a. 



Thus the continued product may be replaced by 



or by e^ 2 ^ 2 * 7 ; a function which is coincident with it when a is 

 infinitesimal. When a is finite both are, as we have seen, in- 

 finitesimal. 



Consequently, 



=-f l du 



*^" J o J n 



(8), 



