J. F. EXCKE ON THE METHOD OF LEAST SQUARES. 331 



is equal to a constant. Any function ,^, _ ^ , can only 



contain, besides this common constant, such members as vary 

 with the value of (M — x), or are a function of (M. — x). But 

 whatever function may be assumed, a sum of products of the form 

 (M — x) f (M— A') will never in general equal = 0, by virtue of the 

 single equation (4.). For let it be granted that it might happen 

 that for the values M, M', ^M" . . . this sum might, with the 

 equation (4.), = 0, still in all cases in which, with the un- 

 changed sum M + M' + M" ... = mp, somewhat different 

 values M — «, M' + «, M" — /3, M"' + /S, &c. have been found, 

 a new equation would arise, which, if the arithmetical mean 

 holds good, must be, together with the equation (4.), = 0. 

 But from the infinite diversity which not only may, but, ac- 

 cording to experience, will be found to exist in the amount of 

 the changes of M, M', M", as well as in their distribution, there 

 can be no function which shall fulfill all these conditions at 

 once. Although the values of M— jo, M' — p, M"— j9 are not ab- 

 solutely independent of each other, because p depends on their 

 sum, yet they must, in case the arithmetical mean holds uni- 

 versally good, be considered as independent variables, because 

 the only equation which expresses this dependence, with every 

 number of observations, disappears in consequence of the in- 

 finite diversity of the values which may still be found after this 

 equation has been fvdfiUed. 

 This equation, 



i;} A _dlog. (^ A) _ , 

 A ~ A d A ~ ' 



where k is an arbitrary constant, gives immediately the form of 

 $ A. Integrating, 



Const. + log. <p A = ^ k A"; 



or, <^ A = X e ^ J 



in which formula the value of the constants remains still be to 

 be determined. 



In regard to k, the above remark, that <f A must be a maxi- 

 mum for A = 0, shows at once that k must always be nega- 

 live. It may therefore be more convenient to write 



9 A = X e 

 z 2 



