126 J. TT. (rihh$ — Equilibrium of ffetero(jfefieous Substances. 



Xow, using Lagrange's ''■method of multipliers,"* we will sul)- 

 tract 7' {:^ 6rf + I^ Dr>) -P(2:'o\' -\- 2: I)r) from the first member 

 of the general condition of equilibrium (^H), 7' and P being constants 

 of which the value is as vet arbitrary. We might proceed in the 

 same way with the remaining equations of condition, but we may 

 obtain the same result more simply in another way. We will first 

 observe that 



+ (:i" 6m„ + >: Dm„) S„ = 0, (42) 

 which equation would hold identically for any possible values of the 

 quantities in the parentheses, if for r of the letters 3j, 3^, . . . ^„ were 

 substituted their values in terms of the others as derived from equa- 

 tions (38). (Although 2 ,, Sg^ . . . 3n do not represent abstract quanti- 

 ties, yet the operations necessary for the reduction of linear equations 

 are evidently applicable to eqiuitious (38).) Therefore, equation (42) 

 will hold true if for 3^, Sg, . . . 2„ we substitute n numbers which 

 satisfy equations (38). Let 3/,, J/j, . . . 3I„ be such numbers, i. e., 

 let 



^»j J/j + bo 3I2 . . . + b^ J/„ = 0, '^ r equations, (43) 

 etc. ) 



then 



J/j {:^Sm^^ :^Dm^) + M2 {2 6m2-h2Dm2) . . . 



+ J/„ {:S 6m„ + 2i' Din„) = 0. (44) 



This expression, in which the values of « — r of the constants J/,, J/g? 

 . . . JI„ are still arbitrary, we will also subtract from the first mem- 

 ber of the general condition of equilibrium (37), which will then 

 become 



2D€+ 2 {t d>;) - :^ (/) dv) -{- 2 (;/ ,6m,) . . + 2: (//„ 6m„) 



- T2 d// + 1^2 6v - M, 2 6m , . . . + J/'„ v (^m„ 



- T2Dr^-\-F:SDv -3/, :2Dm, .. . -J/„ >Z)w„^0. (45) 



That is, having assigned to T, P, Jl,, JJ^, . . . 3/„ any values con- 

 sistent with (43), we may assert that it is necessaiy and sufficient for 

 equilibrium that (45) shall hold true for any variations in the state 

 of the system consistent with the equations of condition (39), (40), 

 (41). But it will always be possible, in case of equilibrium, to assign 

 such values to T, P, M,^ Jf^, . . M^, without violating equations (43), 



* On account of the sign ^ in (37), and because some of the variations are incapable 

 of negative values, the successive steps in the reasoning vriU be developed at greater 

 length than would be otherwise necessary. 



