Reducing Observations relating to Several Quantities. 187 



: It may be worth while showing that, not only is the inde- 

 terminateness in question rare and remediable, but it is also 

 apt to be slight and neglectible. For it tends to be of an 

 order which is insignificant in comparison with the "probable 

 error to which the solution is liable. It will be sufficient to 

 show that, in the case of a single qusesitum and observations 

 obeying the simplest law — ranged under one and the same 

 probability-curve — as the number of observations increases, 

 the space which is left blank at the centre tends to become 

 indefinitely small in comparison with the probable error of the 

 solution by the Method of Least Squares. The Modulus of 



that error is —^= (where c is the Modulus of the primary Pro- 



bability-curve) . Now consider the probability that a certain 



fraction of that Modulus, say i —7=-, measured from the centre 



v n 

 (the real point supposed known) will have remained blank. 

 The probability that any single observation should fall outside 



the space +i—y=- may be written ( 1 — 6(i — ^=j j, where 6 is 



the integral of the error-function. Hence the probability that 

 all the observations should lie outside that limit is 



M^J)" 



Expanding and taking logarithms, we see that the logarithm 

 of this probability is of the order — \/ n ; that is, it becomes 

 indefinitely improbable that the iih part of the probable error 

 should be left indeterminate. This investigation may be 

 extended to prove the required proposition in its generality. 



curve whose centre is the Arithmetic Mean of the given set (and whose 

 Modulus is the — pth part of the Modulus appertaining to the given ob- 

 servations). If, now, we must put one point as representative of all the 

 series of points which the source may assume, the best representative, that 

 which minimises the detriment incident to inevitable error, is the centre 

 of the curve of sources; that is, the Arithmetic Mean of the given observa- 

 tions. Now had the curve of sources given by Inverse Probability con- 

 sisted of a horizontal line at the centre, as in the case before us, the reasoning 

 by which the central point is judged best would not have been affected. 



h 

 I do not forget that, when the law of error is other than y= „€-**, this 



reasoning is applicable less directly, and only in virtue of the explanations 

 referred to in the third note to p. 184. 



