212 Mb. ELLIS, ON THE METHOD OF LEAST SQUARES. 



^'' .V = (\, o, + X„ 6, + &c.) -!, + (X, a, + \,b,+ &c.) -| + &c. 



that is to say, as is seen on referring to (B), 



.V = jUi Fi + /uj Fj + ... + n„ V„, 

 as before ; which proves that the system (/3') gives the same value for x as the most advantageou.i 

 system of factors. Moreover, as (/3') is symmetrical in x and a, y and 6, &c. it is clear that it 

 will also give the most advantageous values for y and the other unknown quantities. 



When the law of probability of error is the same at every observation k^ - k^= Sec. and (/3') 

 reduces itself to (/3) given at p. 208 as the result of the method of least squares. In the general 

 case, it expresses the modification which the method of least squares must undergo, when all 

 the observations are not of the same kind, namely, that instead of making the function 

 2 {ax + by + &c. - F)" a minimum with respect to xy, &c., we must substitute for it the 



function 2— (ax + by + &c. - Vf, and then proceed as before. 

 At 



Such, in effect, is Laplace's demonstration, except that he supposed the law of error the 

 same at each observation. The form in which I have presented it is wholly unlike his. The 

 introduction of Fourier's theorem enables us to avoid the theory of combinations, and also the 

 use of imaginary symbols. It must be admitted that there are few mathematical investigations 

 less inviting than the fourth chapter of the Theorie des Probabilites, which is that in which the 

 method of least squares is proved. 



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



_ + I X ,+ * -+00 



P=- f du f da I rfe, ... / rfe„ ^, e, ...0„ f„cosa (i* - S^e)- 



It is certain that 2/ue lies between the limits ± c;. Therefore when I = ta, P should be 

 equal to unity. I proceed to show that this is the case. 



p =- e""''" dti / da de,... J de„ 0, t, ... ^,6,. cosa (?< - 2/it) 



when OT = . 



Effecting the integration for u. 



P 



c 



when m = 0, 



= y= e *'"'da / rffi ... / d6„0,6i ... ^„e„ cos a S«€ ... (10) 



r 



'" "'cos atidu = e ' 



m 



, + "> 

 and / e " "'^"' sin au du = 0. 



Integrating for a, we see that when m = 



P^ = /^"de, ... [*' de.<p,e, ... 0,e, e -'■"-""'... (ll). 

 Or, 



P^ = J </),«, rfe, ... / <p„e„de (12). 



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



P = 1, which was to be proved. 



