THE METHOD OF MEANS. 419 



quantities, as before, their reciprocals are and , the 



a o 



mean of which is J ( + y), and the reciprocal again is 



-. Other kinds of means might no doubt be invented 

 for particular purposes, and we might apply the term, as 



De Morgan pointed out e , 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 themselves will make no alteration in 

 the value of the function. Symbolically, if <p (y, y, y . . . .) 

 = (f> ( x l9 x 2 , x 3 . . . .), then y is a kind of mean of the 

 quantities x l9 x 2 , &c. 



The geometric mean is necessarily adopted in certain 

 cases. Thus 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 geo- 

 metric mean between the forces at the. beginning and end 

 of the path f . 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 of the one 

 body (see p. 410). 



In almost all the calculations of statistics and commerce 

 the geometric mean ought, strictly speaking, to 'be used. 

 Thus if a commodity rises in price 100 per cent, and 

 another remains unaltered, the mean rise of 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 A/I -00x2-00 

 or i to 1*41. The difference between the three kinds of 

 mean in such a case, as I have elsewhere shown , is very 

 considerable, being as follows 



e 'Penny Cyclopaedia/ art. Mean. 



f Thomson and Tait, ' Treatise on Natural Philosophy/ vol. i. p. 366. 

 K ' Journal of the Statistical Society/ June 865, vol. xxviii. p. 296. 



E e 2 



