1879.] 



the IjCtiv of the Geometric Mean. 



371 



at x to be -. Then the ordinate at x is proportional to the (in- 



x 



finitesimal) probability of x, and the mass of the section bounded by 

 the ordinates at x^, x 2 measures the probability of a measure lying- 

 bet ween x 1 and x%. If we are dealing with a very large number of 

 measures, this mass will be found to represent the number of measures 

 which actually lie between those limits. The mass of the lamina is 

 bisected by the ordinate at x = l (fig. 2). 



Fig. 2. 



[y\/ 7r = Aexp. ( — A 2 log 2 #). 



1 density at w = ~ . 

 I x 



2°. Regard the equation yV ' tt—Ii exp. ( — h 2 \og~x) as that of a 

 cylinder. Let it be cut by another cylinder zx=l, at right angles to 

 it. The solid included between these surfaces and the planes xz, xy, 

 obviously corresponds to the lamina of 1°, volume being read for mass 

 (fig. 3). Methods 1° and 2° thus represent in one figure both the fre- 

 quency and the facility functions. 



Fig. 3. 



{y\/n = 7i exp. ( — h 2 log 2 x) 

 -4 



