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



The mean value of e' is / t'cpede = 2 k'. 



Hence, that of S^'e' is aS^^fc'. 



The mean value of liM/^ieie.^ is zero, positive and negative errors of the same magnitude 

 occurring with equal frequency on the long run. 

 Consequently, 



mean of (2Me)= = 2 2,.x-fc' (14); 



and, therefore, as before, 2m^^" is to be made a minimum. The rest of the investigation is of 

 course the same as that of Laplace. 



Nothing can be simpler or more satisfactory than this demonstration. It is free from all 

 analytical difficulty, and applicable whatever be the number of observations, whereas that of 

 Laplace requires this number to be very large. 



Recurring to equation (ll), differentiating it for m\ and then making m = o, we find 



f pti^du= f ^t, rfe, ... f (pe„de„('2.ney = 'i'S./j.'k^ ; 



and as the first member of this equation is evidently the mean value of ^^- or of (2^£)S this is a 

 new verification of our analysis. 



As an illustration of Gauss's principle, let the fourth power of the error be taken as the 

 measure of its importance ; 



(2/jie)* = 2m*«' + 6 2,u,-/V £,'63" + terms involving odd powers of e. 

 Therefore, 



mean of (S/xe)' = 2 2mV + 24>-2^{',i%k^^k2^ (15) 



and /Ui ... n„ must be so determined that this may be a minimum. 



I have already said that the results- given by what Laplace called the most advantageous 

 system of factors are not strictly speaking the most probable of all possible results. 



As the distinction involved in this remark seems to me to be essential to a right apprehension 

 of the subject, I will endeavour to illustrate it more fully. 



Recurring to the equations of condition, as given in p. 208, we see that the values Laplace 

 assigns to the factors ^^ ^2 &C', are independent of Fj V^ &.c. They depend merely on the 

 coefficients a b &c., which are quantities known a priori, i. e. before observation has assigned 

 certain more or less accurate values to the magnitudes F, F, &c. All we then can say is, that 

 if we employ Laplace's system of factors, and also any other, in a large number of cases (the 

 coefficients a b &c., being the same in all) we shall be right within certain limits in a larger 

 proportion of cases when the former system of factors is made use of than when we employ the 

 latter. And this conclusion is wholly irrespective of the values of Fj F^ &c., and consequently 

 of those which we are led in each particular case to assign to ir y &c. The comparison is one 

 of methods, and not at all one of results. But when F, V^ kc. are known, another way of 

 considering any particular case presents itself. We can then compare the probability of different 

 results. For, let us consider a large number of sets of equations of condition (in each of which 

 not only are a b &c. equal, as in the former case, but also F, Fj &c.) The true values of 

 the elements x y &c. may be different in each. But in affirming that ^ »; &c., are the most 

 probable values of a: y &c., we affirm that the true values of x y &c. are more frequently equal 

 to ? >; &c. than to any other quantities whatever. Here we have no concern with the method 

 by which the values ^ t) &c. were obtained. The comparison is merely one of results. 



As for one particular law of error (that considered in p. 206), the results of the method of 

 least squares are the most probable possible ; and as the function by which this law of error is 



