132 



be 2) we may safely adopt the results. If there is a 

 sensible différence we may repeat the process by the 

 introduction of still smaller intervais. 



Remark IL For those values of x for which the fre- 

 quencies are very small — such as is usually the case 

 towards the limits — the values of the z but particularly 



those of , are of necessity unreliable. If we want to 



z 



know the degree of this unreliability we may change the 

 given frequencies by amounts such as in our judgment 

 may well subsist in our numbers, owing to incertainties 

 of the measures, scarcity of data or other causes. So, for 

 instance: if in table 2 we change the frequencies 0.001 

 and 0.006 between the values x = 0.0 and 0.5 resp. 0.5 



and 1.0 to 0.000 and 0.007. the two first values of ^ 



z 



viz. 6.7 and 1.18 will change to oo and 1.0. 



An equal change in the considérable frequencies between 



X = 2.0 and 3.0 will hardly affect the values of \ at 



z 



ail. Whereas the reaction curve, therefore, is very reliable 

 for the middie values x — 1.5 to x = 3.0 it is enormously 

 less so towards the extremities of the curve. 



Remark IIL Suppose that for a certain very small 

 increase of x, the z did not change at ail. The case 

 would necessarily occur if the ordinales of the z curve 

 were partly increasing with the x, partly decreasing. We 

 would then hâve for the middie of the interval z^ = 0.000 



consequently , = oo. The reaction would thus be infinité. 



As such a thing cannot exist in nature, the supposed case 

 cannot présent itself. 



15. Hxamples. From table 2 it will be seen how very 

 easy the computation of both the normally spread func- 



1 , . 0-1 1 u 



tion z and the reaction-curve --,- turns out to be. Havmg 



