in the Calculation of Errors. 105 



above the point x— 1 had better be taken in ; and accordingly, 

 if they are omitted, the modulus of the weighted mean 

 is greater than if they were taken in. But, if the same 

 weighted observations' are incorporated with two averages, 

 one of which has a larger modulus than the other, the limit 

 at which it will cease to be advantageous to take in a new 

 observation will be reached sooner in the case of the smaller 

 modulus. Thus, as we move inwards towards the centre, we 

 shall reach u sooner than u' ; in other words, u' is an inferior 

 limit to u. 



To determine m / , in the absence of tabulated integrals for the 



function -^e x \ I have adopted a method which is in fact 



x 



more appropriate to the case in hand — where the observations 

 are broken up into little bundles : — I have plotted the ordinate 



of the curve y — — 2 e° 2 for the points x = l, *9, '8, *7, &c; and, 

 oo 



joining their tops, observed the point at which yX (1—x) 



begins to be greater than twice the area contained between 



the ordinate at the point x, the abscissa, the ordinate at the 



point 1, and the locus joining the tops of the ordinates. This 



limit proves to be about *3. Accordingly at least all the 



observations below this value of x are to be rejected. 



By similarly operating on the observations above the point 

 x = l, I find for v f , a limit superior to v, the point x = 2 ; and 

 conclude that at least all the observations above that point are 

 to be rejected. 



(7) A third source of error affecting the computation 

 arises from the imperfect graduation of our instruments and 

 senses, by which we are compelled to put for the true value 

 of any object measured a value in the neighbourhood equal 

 to an integer number of degrees, e. g., tenths of an inch. 

 Assuming that the difference between the apparent and 

 real value is more likely to fall short of than to exceed half a 

 degree, and is very unlikely to exceed a whole degree, — the 

 modulus of the error from this source affecting each observa- 

 tion is much less than a degree ; and is therefore small, if the 

 degree is small, as in the case of stature, if the degree be a 

 tenth of an inch ; the unit in which x is measured being 

 3*6 inches. In the case of the cubit the unit is smaller, the 

 error from source 7 greater. 



In computing the errors investigated above, and for other 

 problems, the following notation will be convenient. Let the 

 symbol plus written in full [and similarly minus'] connecting 

 probable errors or moduli denote the cumulation of the errors, 



