EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 235 



where the symbols j/ 8(1) , F 2(1) , etc., are used for greater distinctness to 

 denote the values of q a , F 2 , etc., as determined by a dividing surface 

 placed so that F^O. Now we may consider all the differentials in 

 the second member of this equation as independent, without violating 

 the condition that the surface shall remain plane, i.e., that dp' = dp". 

 This appears at once from the values of dp' and dp" given by equation 

 (98). Moreover, as has already been observed, when the fundamental 

 equations of the two homogeneous masses are known, the equation 

 p'=p" affords a relation between the quantities t, fa, fJL 2 , etc. Hence, 

 when the value of o- is also known for plane surfaces in terms of 

 t, fa, yu 2 , etc., we can eliminate fa from this expression by means of 

 the relation derived from the equality of pressures, and obtain the 

 value of a for plane surfaces in terms of t, /* 2 , /i 3 , etc. From this, 

 by differentiation, we may obtain directly the values of rj &(l) , r 2 (D, T 3(l) , 

 etc., in terms of t, // 2 , /* 3 , etc. This would be a convenient form of 

 the fundamental equation. But, if the elimination of p', p", and fa 

 from the finite equations presents algebraic difficulties, we can in all 

 cases easily eliminate dp', dp", dfa from the corresponding differential 

 equations and thus obtain a differential equation from which the 

 values of ^ S(1) , F 2 (i), F 3(1 ), etc., in terms of t, fa, // 2 , etc., may be at once 

 obtained by comparison with (514).* 



* If liquid mercury meets the mixed vapors of water and mercury in a plane surface, 

 and we use /^ and ^ to denote the potentials of mercury and water respectively, and 

 place the dividing surface so that I\ = 0, i.e., so that the total quantity of mercury is 

 the same as if the liquid mercury reached this surface on one side and the mercury 

 vapor on the other without change of density on either side, then F 2 (i) will represent 

 the amount of water in the vicinity of this surface, per unit of surface, above that which 

 there would be, if the water- vapor just reached the surface without change of density, 

 and this quantity (which we may call the quantity of water condensed upon the surface 

 of the mercury) will be determined by the equation 



do- 



(In this differential coefficient as well as the following, the temperature is supposed to 

 remain constant and the surface of discontinuity plane. Practically, the latter condition 

 may be regarded as fulfilled in the case of any ordinary curvatures. ) 



If the pressure in the mixed vapors conforms to the law of Dalton (see pp. 155, 157), 

 we shall have for constant temperature 



where p z denotes the part of the pressure in the vapor due to the water- vapor, and y 2 

 the density of the water- vapor. Hence we obtain 



d<r 



For temperatures below 100 centigrade, this will certainly be accurate, since the 

 pressure due to the vapor of mercury may be neglected. 



The value of <r for p 2 =0 and the temperature of 20 centigrade must be nearly the 

 same as the superficial tension of mercury in contact with air, or 55*03 grammes per 

 linear meter according to Quincke (Pogg. Ann., Bd. 139, p. 27). The value of <r at 

 the same temperature, when the condensed water begins to have the properties of water 



