EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 113 



Let us give to r\ or t, to m l or fjL v ...to m n _ 1 or /z n _ 1} and to v, 

 the constant values indicated by these letters when accented. Then 



by (165) 



(177) 



Now *-< 



approximately, the differential coefficient being interpreted in accord- 

 ance with the above assignment of constant values to certain variables, 

 and its value being determined for the phase to which the accented 

 letters refer. Therefore, 



'.-<)** (179) 



and * = i (m "- 7n " /)2 - (180) 



The quantities neglected in the last equation are evidently of the 

 same order as (w n w n ') s . Now this value of $ will of course be 

 different (the differential coefficient having a different meaning) 

 according as we have made i\ or t constant, and according as we have 

 made m x or ^ constant, etc. ; but since, within the limits of stability, 

 the value of <3>, for any constant values of m n and v, will be the least 

 when t, p, /*!> /*-! have the values indicated by accenting these 

 letters, the value of the differential coefficient will be at least as small 

 when we give these variables these constant values, as when we 

 adopt any other of the suppositions mentioned above in regard to 

 the quantities remaining constant. And in all these relations we 

 may interchange in any way T/, m p . . . m n if we interchange in the 

 same way t, [t v ... fJL n . It follows that, within the limits of stability, 

 when we choose for any one of the differential coefficients 



drf dm l '"'dm n 



the quantities following the sign d in the numerators of the others 

 together with v as those which are to remain constant in differen- 

 tiation, the value of the differential coefficient as thus determined 

 will be at least as small as when one or more of the constants in 

 differentiation are taken from the denominators, one being still taken 

 from each fraction, and v as before being constant. 



Now we have seen that none of these differential coefficients, as 

 determined in any of these ways, can have a negative value within 

 the limit of stability, and that some of them must have the value zero 

 at that limit. Therefore in virtue of the relations just established, 



one at least of these differential coefficients determined by considering 

 G. i. H 



