1882.] 



fluctuations in a variable. 



189 



the corresponding ordinates of the curves PJl$ v P 2 R 2 Q 2 . Suppose 

 N middle point of LM. Then EN is the mean of the magnitudes 



PL, QM, but RjV, R 2 N, are respectively less and greater than 

 the means of PJj, Q^I, and of P 2 L, Q 2 M. 



It is evident from the figure that in a curve for which 



ay 



dx* 



is of constant sign, the mean of any pair of ordinates is greater 



or less than the ordinate from which they are equi-distant on each 



d 2 f ... 

 side, according as -^ is positive or negative. Therefore for any 



CLOG 



portion of such curve, the mean value of the ordinate is greater 

 or less than the ordinate corresponding to the mean abscissa as 



d\f 



dx 



5 is positive or negative. I do not propose to consider cases 



d 2 f 



in which ~ changes sign between the limits. 



Recurring to the figure, since y represents the ordinate of the 

 line PR Q, -~ is constant. 



Therefore &-& W 



lneretore d^~ df\dx) ' 



whence it follows that we may use , - a as our test ; and say that the 



