1879.] 



the Law of the Geometric Mean. 



369 



Arithmetic Mean, and the formula} of the ordinary theory are at once 

 applicable to it, the weight ascribed to it being h, as above defined. 



When the ^/-series is so treated, we can at once pass from it to the 

 ^-series. Thus the probability, frequency, &c, of any x are the same 

 as for the corresponding y : and any term interpolated in the ?/-series 

 will be the logarithm of the corresponding term in the ^-series. 



1. If it be granted that the Geometric Mean of the measures is the 

 most probable value of the quantity measured, it follows that measures 

 which differ from the mean have a probability which becomes less 

 as the discrepancy becomes greater, either in excess or defect. We 

 naturally seek for some scale which shall define the respective proba- 

 bility to be assigned to each measure of any given series. 



Thus, if x be one of the measures, and a be the Geometric Mean of the 

 whole series, and if we can assign a form to the function 0, such that 



$(~^ ^ s ^ ne probability of x, the curve y=4> will afford us (gra- 

 phically) the required scale, and 0^-^will be the "Law of Frequency." 



2. By a method analogous to the first method of Gauss (" Theoria 

 Motus," ii, 3, 186), it can be shown that 



00=B exp. (-fi2 l og 2 ^=Be~ h <^- a )\ 



where B and h are constants. B is determined from the condition 

 that the sum of the probabilities of all the measures must be unity ; h 

 depends on the grouping of the measures, being large when they agree 

 closely and small when they are generally discordant ; h is thus taken 

 as a measure of the precision of the series in general, and is briefly 

 called the "weight" of the series, or sometimes the "weight" of the 

 mean derived therefrom. (See fig. 1.) 



Fig. 1. 



# = B exp. ( — h 2 log 2 - ). 



\ at 



S. In forming a scale of probability, 0^-^> which shall be appli 



