1879.] 



the Law of the Geometric Mean. 



373 



by a known approximation, the chance ultimately takes the form 

 / — exp. ( — — \ If we are asked the chance of a ratio x, we put 

 x=a. r , and thus find log x—r log a ; the chance required is then 



"J nir \ w Vlog a,) /' 



corresponding to the law of frequency. 



Next determining the chance that the estimate shall lie between 

 narrow limits x and x+x, we find that we have to multiply the above 

 expression by x'-i-xloga. Thus, if we take &~ 3 =w(loga) 3 , the ex- 

 pression agrees exactly with that already found for the law of facility. 

 This verification is interesting, for though it depends on certain 

 special suppositions, the process seems to throw light on the genesis of 

 the law, and the significance of the modulus h. 



8. Another method of exhibiting the law, suggested by Mr. Galton's 

 Method of Intercomparison, is the following. Let the series of 

 measures be represented by a series of ordinates : arrange these side 

 by side at equal small distances and in order of magnitude. Their 

 extremities will then lie on a curve of contrary flexure, which Mr. 

 Galton calls an Ogive ; we may speak of it as the " curve of distribu- 

 tion" Its equation can be shown to be 



hJcx 



— \ e ^ dt=evi. (hlogy), 

 Jo 



h being a constant depending on the interval between the ordinates as 

 they stand (fig. 5). 



Fig. 5. 



kkw = erf. (h log >/) . 



OB = OB' = K)A. 

 BC=B , C / = iOA. 



