26 ON THE METHOD OF LEAST SQUARES. 



less inviting than the fourth chapter of the Theorie des Proba- 

 bilites, which is that in which the method of least squares is 

 proved. 



It may be worth while to recur to the general formula: 



j r+l r 00 /+ /+ 



P = -I du I da. \ de l . .. I de n ^ e 1 ...(f> n e n cos a (u 



TTjji Jo' >_ J-oo 



It is certain that 2/^e lies between the limits + co . There- 

 fore when I = co , P should be equal to unity. I proceed to show 

 that this is the case. 



doi f de,...! Je n ^, 1 e 1 . 



J-oo '-w 



P =- 



when m 0. 



Effecting the integration for M, 



P L = 



1 f 00 _ * 2 / %+ ' r + * 



I e~4^2 j a I J 6i0>% de n <j>^ ... </> n e n cos a 2/*e ...... (10) 



VTT * *- '- 



wr 

 when m 0, 



since 



/ " f VTT. 



/+ 



e- m * 



/_ 



r +o 



and I e~ m2u2 sin ocw du - 0. 



^ -00 



Integrating for a, we see that when m = 

 P. = f" *,... f"^^...^^ 



^ -00 / -00 



Or, 



And as each of these integrals is separately equal to unity, 

 P = 1, which was to be proved. 



