EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 81 



In regard to the relation of the potential p x to the potentials occurring 

 in equation (58) it will be observed, that as we have by integration 

 of (58) and (59) 



eWfl'-pV+^'m.'.. . + //>;, (60) 



and eWfl'-pV+^'m,,'; (61) 



therefore fJL x 'm x ' = // a 'm a '- + /V m </- ( 62 ) 



Now, if the fluid has besides 8 a ,...S g and S h ,...S k the actual 



components $ . . . S n , we may write for the fluid 



Se" = tf' 8n" -p"8v" + yu "<5m fl ". . . + H g "$m ff " 



+ V h ''6m h '\.. + v k ''Sm k ''+fi{'8m{'... + v n ''8m n '', (63) 



and as by supposition 



m;S> z = m a '@ a ...+m;@, (64) 



equations (43), (50), and (51) will give in this case on elimination of 

 the constants T, P, etc., 



t'=r, P '=rr, (65). 



and m x 'fjL x ' = ra a > a "- . . + m////'. (66) 



Equations (65) and (66) may be regarded as expressing the conditions 

 of equilibrium between the solid and the fluid. The last condition 

 may also, in virtue of (62), be expressed by the equation 



m a >; ... 4- m//V = m a > a ". . , + m,X". (67) 



But if condition (53) holds true of all bodies which can be formed 

 of S a , ... S g , S h ,... S k , S h ... S n , we may write for all such bodies 



- t"ti +p"v - fji a "m a ... - fjL g "m g - /VX 



... - /// 'm u - ^"m, ... - pS'm* ^ 0. (68) 



(In applying this formula to various bodies, it is to be observed that 

 only the values of the unaccented letters are to be determined by 

 the different bodies to which it is applied, the values of the accented 

 letters being already determined by the given fluid.) Now, by (60), 

 (65), and (67), the value of the first member of this condition is zero 

 when applied to the solid in its given state. As the condition must 

 hold true of a body differing infinitesimally from the solid, we shall 

 have 



,; ...- ft," dm,' 



J ... - p t "dm h ' ^ 0, (69) 



or, by equations (58) and (65), 



(//; - /z a ") dm a '. . . + (/V - O dm; 

 +(/V-^ // )^V...+( / / ; ;- y u/)dm;^0. (70) 



Therefore, as these differentials are all independent, 



*,' =//.",.../;.; = //;', ti /",... ft' Sft"; (71) 



G.I. F 



