Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 207 



Consequently, we ai'e thus led to adopt one particular law of probability of error as alone 

 congruent with the rule of the arithmetical mean. 



But, in fact, we are perfectly sure that in different classes of observations the law of proba- 

 bility of error must vary, and we have no direct proof that in any class it coincides with the 

 form assigned to it. Therefore one of two things must be true, either the rule of the arith- 

 metical mean rests on a mere illusory prejudice, or, if it has a valid foundation, the reasoning 

 now stated must be incorrect. Either alternative is opposed to Gauss's investigation. For the 

 reasons already given, we are, I think, led to adopt the latter, and then the question arises, wherein 

 does the incorrectness of the reasoning reside ? It resides in the ambiguity of the words )iioiit 

 probable. For let us consider what they imply in the theory of probabilities a posteriori. 



Suppose there were m different magnitudes a, a., ... o„,, and that each of these were observed 

 n times in succession. Let this process be repeated p times, p being- a large number which 

 increases sine limite. Thus we shall have pm sets of observations each containing n obsei-vations. 



Of these a certain number K will coincide with the set of observations supposed to be actuallv 

 under discussion ; and we shall have tiie equation. 



ki + k,j + ... k„ = K : 



where k is that portion of K which is derived from observations of Oj.. 



Then, ultimately, the most probable value which the given series of observations leads us to 

 assign too, is (supposing a is susceptible only of the values 0,0,... «,„) equal to a,, r being 

 such tliat the corresponding quantity k, is the maximum value of k. 



To make the case now stated entirely coincident with the one which we are in the habit of 

 considering, we have only to suppose (making m infinite) that the series of magnitudes 

 a, ... a,„ includes all possible magnitudes from -co to -1- 05 . 



Now from this statement, it is clear there is no reason for supposing that because the 

 arithmetical mean would give the true result if the number of observations were increased 

 sine limite, it must give the most probable result the number of observations being finite. 



The two notions are heterogeneous : the conditions implied by the one may be fulfilled 

 without introducing those required by the other : and we have already seen that by losing sight 

 of this distinction, we are led to the inadmissible conclusion, that a principle recognised as true 

 a priori necessarily implies a result, viz. the universal existence of a special law of error, not only 

 not true a priori, but not true at all. 



Having stated what seem to me to be the objections in point of logical accuracy to this 

 mode of considering the subject, I will briefly point out the manner in which, from tlie law of 

 error already obtained, the method of least squares is to be deduced. 



Let 



6, = 0|.r + 6,2/ + &c. - r, 

 fa = a.^x + h^y + &c. - V^ 



( 



&c. = &c. I 



be the system of equations of condition, which are to be combined together so as to give the 

 values of x, y, &c. The error committed at the first observation is 61, at the second tj' ""'' 

 so on ; each observation corresponding to an equation of condition. 



The probability of the concurrence of all these errors is, (according to the law of error 

 already arrived at) proportional to 



^- h'[iii,x * b^y + tie. - I'l)' + (oax + b,ij + - Ts)' + &c.] 



and it is to be made a maximum by the most probable values of ,r, y, &c. These values will 

 therefore make 



