370 



Mr. D. McAlister on 



[Nov. 20, 



cable to any series of measures of which the mean and the weight are 

 known, we must suppose the value of to be given (either graphi- 

 cally or by table) for all possible values of x. In other words, we 

 must suppose the number of entries to be indefinitely great and 

 indefinitely close together. It is clear that the actual probability of 

 any individual measure becomes in such a series infinitesimal, though 

 the relative magnitudes of any pair of probabilities remain unaltered. 

 This is indicated in the formula of §2 by the constant B becoming 

 infinitesimal. To avoid this inconvenience we are led to modify our 

 original problem, and ask, not " what is the probability of a measure 

 x?" but "what is the probability of a measure lying between the 

 very close limits x and x + hx ? " 



4. From the principle indicated above, viz., that on the whole there 

 are as many measures less than the mean as there are greater than 

 the mean, we deduce that the required probability is 



* exp. ( - 7i 2 log 2 

 V 7T \ a/ x 



the constant factor h-r- *J ir results from the condition that the 

 sum of all the values of this function from x=0 to x—cc must be 

 unity. And it will be found that 



p *-«p.(-»iog«?)*-i. 



Jo Vyj. \ aj x 



as should be the case. 



The coefficient of 8x, viz.: — h exp. ^— 7i~ log 2 v 7 ? 7 " • a '> I have 



ventured to distinguish from the previous function by calling it the 

 " Law of Facility" This function may be defined in words as "the 

 ratio which the chance that a measure lies within a given small 

 interval bears to the magnitude of the interval. 



5. Tracing the curve which represents the " Law of Facility," we 

 find that its form resembles that of the curve of frequency, but that 



the maximum ordinate occurs at x = a exp. ^ — cjj^) * n ^ e ^ ormer > anc ^ 



of course at x=a in the latter. The latter result implies that the 

 mean is the most probable value : the former means that there is more 

 chance of a measure lying within a given small fraction (ca?) of 



a . exp. ^ — 2h?) ^ an w *^ n ^ e same small fraction of any other value 



that can be named. 



6. We may graphically represent our results in three ways : — ■ 



1°. Trace the curve y^/ 7r=h exp. ( — /rlog 2 a?), (the mean being- 

 taken as unity) . Leu the space between the curve and Ox be the boundary 

 of a lamina of varying density. Take the density along the ordinate 



