1879.] 



the Law of the Geometric Mean. 



375 



The gauges of the series given by the weight and the quartiles and 

 octiles will generally suffice. 



11. Let two fallible measures be given whose respective weights 

 are known, and let their product be formed. This product will be a 

 fallible measure like its factors. It can be shown that the product is 

 subject to a law of facility of the form already obtained, and further 

 that, as in the ordinary theory, the weight is such that the square of 

 its reciprocal is equal to the sum of the squared reciprocals of the 

 weights of the factors. More generally, if x h x 2 . . . be a series of 

 measures whose weights are hi, h^ • . . , and if z be connected 

 with them by the equation 



z=x 1 "i X aj 3 »« x . . . ., 



Then h the weight of z is given by 



These results are strict ; that which follows is closely approximate 

 and applies to series whose weight is not small. 



Let / (a x , a% . . . ) be any function of the fallible measures 

 ■c&i, <x 3 . . . , each of which may be the mean of some considerable 

 number of measures, so that the weights are respectively hi, h 2 . . 

 Required the weight to be assigned to the value of the function 

 obtained from these fallible values of its variables. We are able to 

 show that if hf be the weight in question 



i.-( »iog/ yi + ( »og/ yi + .... 



7i/ 3 \S log a Y ) h^ \h log a J h 2 2 



12. It only remains to establish the practical method presented in 

 § 1. We have 



Chance of a logarithm 1 _ / Chance of a measure 

 lying between y and y + cy J 1 lying between x and x + Bx 



_ h / x\ cx_ h -2 



" vV ex PV /i4 °^ fl/i" A ex P- (-hhj-ioga ) cy. 



This is the expression of the ordinary law of grouping of the ?/'s 

 round their arithmetic mean (log a) . 



It follows that, for example, the number of #'s between x r and x s 

 must be proportional to 



4-f ! '*exp.(-AV )%: 



V 7T Jyr 



this can be readily' evaluated by means of the ordinary tables of the 

 error-function. 



Again, still using h for the weight, the " probable- error " of the 



