ON THE METHOD OF LEAST SQUARES. 17 



observations the law of probability 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 arithmetical 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 most probable. For let 

 us consider what they imply in the theory of probabilities 

 cl posteriori. 



Suppose there were m different magnitudes 1 2 ...a m , 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 observa- 

 tions each containing n observations. 



Of these a certain number K will coincide with the set of 

 observations supposed to be actually under discussion ; and we 

 shall have the equation 



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

 of %. 



Then, ultimately, the most probable value which the given 

 series of observations leads us to assign to a, is (supposing a is 

 susceptible only of the values a^...^) equal to a r , r being 

 such that the corresponding quantity k r 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 m includes all possible magnitudes from GO to + co . 



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. 



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