88 BUTLER 



ART. D 



represents the rate of increase of f with the quantity of the 

 component S\, when the temperature, pressure and quantities 

 of the other components remain constant. It is therefore the 

 'partial free energy of the first component. According to equa- 

 tions (44) and (45), ^i is also given by 



Ml = (jt) ' (46) [104] 



and by 



Ml = f T^) , (47) [104] 



\afn,\/ 1, V, TOj, . . . m„ 



i.e. /ii is equal to the rate of change of e with mi, when the en- 

 tropy, volume and quantities of the other components remain 

 constant, and to the rate of change of \p with mi, when the 

 temperature, volume and quantities of the other components 

 remain constant. 



Now all the terms in (44) are of the same kind, that is mul- 

 tiples of quantities {t, p, ni, etc.) which depend on the state of 

 the system, by the differentials of quantities (t/, v, mi, etc.) 

 which are directly proportional to the amount of matter in the 

 state considered. We may therefore integrate (44) directly, 

 obtaining: 



e = tr] — pv -\- mmi + n^rrii . . . + Urmn, (48) [93] 



whence by (14), (15) and (16) : 



\p = —pv-\- mrrii + H2ni2 . . . + Unnin, (49) [94] 



f = Mi^i + M2W2 . . . + Hnm„, (50) [96] 



X = tV + MlWl + /I2W2 . . . + Mn^n- (51) [95] 



A concrete picture of the process involved in this integration 

 may be obtained as follows. If we take a homogeneous mass 

 having entropy 7? and volume v, and containing quantities mi, 

 nii, . . . m„ of the components >Si, 82,--. Sn, and add quantities 

 of a mass of the same composition and in the same state; t, p, 

 Mi> M2, etc., all remain unchanged and (44) may be apphed to a 

 finite addition: 



