210 Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 



It is clear this is the greatest value the integral in question can have, and therefore as n increases 

 sine limite, the continued product 



J " 0,£,COS/KjO6]rfei ... /j"0„€„ COS/U„a6„rf€„ 



decreases sine limite, (being the product of n factors each less than unity) except for values of a 

 differing infinitesimally from zero. 



Let k" = J (pe.e'de, «' = Jq (pe.e^de, 



and develope each of the cosines in the above-written continued product. It is thus seen to be 

 equal to 



1 - a'-2^'k'- + a* (\ S^iV + 2m? M2 ^I '4) - &c. 



Again, n being very large and ultimately infinite, it is evident that 2/u''k' is of the same 

 order of magnitude as n, while Sju? M2 ^1 ^2 '^ ^^ ^^^ order of n\ the former term of the coefficient 

 of a' may therefore be neglected in comparison with the latter, which again may be replaced by 

 ■^CEn" k')", 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 



1 -a'-E,x'k"- + - aU-2/^'ky —a''(^ti'ky + &c. 



2 2.3 



or by e""^^'''''; a function which is coincident with it when a is infinitesimal. When « is finite 

 both are, as we have seen, infinitesimal. 

 Consequently, 



P = 



IT 



- f du f " cos au da. e""^"'''^ (S), 



IT •'a -'0 



P-=7-^,1;kv f'e~ ''■''''''' d'' = -7= f'^^^^'^'^'dv (9), 



{■n-Ztx'k ) Jg VT-^O 



where we have supposed 



M = 2 {S.^^k^v. 

 It is evident, that whatever / may be, this expression for P is a maximum when 



'S.fi.'k'' is a minimum. 

 Hence we get the following remarkable conclusion : When the number of observations increases 

 sine limite the most advantageous system of factors are those which make 



lifi'k^ a minimum. 

 It remains to determine /j. from the condition of the minimum taken in connexion with those 

 already stated, viz. 2,ua = 1, 'Z/xb = 0, Sec. = 0. We have 



^k^lj.dn= \ 



^"'^'^ =M (A). 



^bdfjL 



&c. 



Let X|, Xi...\p be indeterminate factors, then we may put 



^iMi = ffliXi + 61X2 + &c. 1 



klfi2= 02X1+62X2 + &c. [ (B). 



&c. = Stc. I 



::[ 



