xvi.J THE METHOD OF MEANS. 361 



But there is also the geometric mean, which is the square 

 root of the product, V x b, or that quantity the loga- 

 rithm of which is the arithmetic mean of the logarithms 

 of the quantities. There is also the harmonic mean, 

 which is the reciprocal of the arithmetic mean of the 

 reciprocals of the quantities. Thus if a and b be the 



quantities, as before, their reciprocals are - and r, the 

 mean of which is ^ f- + ~\ and the reciprocal again is 



7-1 > which is the harmonic mean. Other kinds of 

 a -f- & 



means might no doubt be invented for particular purposes, 

 and we might apply the term, as De Morgan pointed 

 out, 1 to any quantity a function of which is equal to 

 a function of two or more other quantities, and is such 

 that the interchange of these latter quantities among them- 

 selves will make no alteration in the value of the function. 

 Symbolically, if $ (y, y, y . . . .) = (f> (x ly x?, x 3 . . . .), then y 

 is a kind of mean of the quantities, a^, x^, &c. 



The geometric mean is necessarily adopted in certain 

 cases. When we estimate the work done against a force 

 which varies inversely as the square of the distance from a 

 fixed point, the mean force is the geometric mean between 

 the forces at the beginning and end of the path. "When in 

 an imperfect balance, we reverse the weights to eliminate 

 error, the true weight will be the geometric mean of the 

 two apparent weights. In almost all the calculations of 

 statistics and commerce the geometric mean ought, strictly 

 speaking, to be used. If a commodity rises in price 100 

 per cent, and another remains unaltered, the mean rise of 

 a price is not 50 per cent, because the ratio 150 : 200 is 

 not the same as 100 : 150. The mean ratio is as unity to 

 v/i-oo x 2-00 or i to i '4 1. The difference between the 

 three kinds of means in such a case z is very considerable ; 

 while the rise of price estimated by the Arithmetic mean 

 would be 50 per cent, it would be only 41 and 33 per cent, 

 respectively according to the Geometric and Harmonic 



1 Penny Cyclopcedia, art. Mean. 



8 Jevons, Journal of the Statistical Society, June 1865, vol. xxviii 

 p. 296. 



