■T. 7K Crihbs — Fjipdlihrmni of ITeterogeiieous S^ihstatices. H97 



its characteristic properties as a tension witli reference to any ai'bi- 

 trary snrfiice. Considered as a tension, its position is in the snrface 

 wliicli we liave called the surface of tension, and, strictly speaking, 

 nowhere else. The positions of the dividing surface, however, which 

 we shall consider, will not vary from the surface of tension sufficiently 

 to make tliis distinction of any practical importance. 



It is generally possible to place the dividing siirface so that the 

 total quantity of any desired component in the vicinity of the surface 

 of discontinuity shall be the same as if the density of th.at component 

 were uniform on each side quite up to the dividing surface. In other 

 words, we may place the dividing surface so as to make any one of 

 the quantities 7\, F^, etc., vanish. The only exception is with 

 regard to a component which has the same density in the two homo- 

 geneous masses. With regard to a component which has very nearly 

 the same density in the two masses such a location of the dividing 

 sui'face might be objectionable, as the dividing surface might fail to 

 coincide sensibly with the physical surface of discontinuity. Let us 

 suppose that /i' is not equal (nor very nearly equal) to y^", and that 

 the dividing surface is so placed as to make F-^ =z 0. Then equation 

 (508) reduces to 



da = — ?/s(i) <^« — Ad) (^M2 — Ad) (^Ms — etc., (514) 



where the symbols //scd- Ad)? ^tc, are used for greater distinctness 

 to denote the values of t/s, A? etc., as detei-mined by a dividing sur- 

 face placed so that 7"^ = 0. Now we may consider all the differen- 

 tials in the second member of this equation as independent, without 

 violating the condition that the surface shall remain jjlane, 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' :=jij>" affords a relation between the quantities 

 ^> /^i5 ^2-) etc. Hence, when the value of a is also known for plane 

 surfaces in terms of ^, /<i, yWg, etc., we can eliminate yWj from this ex- 

 pression by means of the rehation derived from the equality of pres- 

 sures, and obtain the value of a for plane surfaces in terms of 

 «, //o, /^3, etc. From this, by differentiation, we may obtain directly 

 the values of 7^s(i), A(i)» Ad)? etc., in terms of ?, /z^, /<3, etc. This 

 would be a convenient form of the fundamental equation. But, if the 

 elimination of p', ^>", and //^ from the finite equations presents alge- 

 braic difficulties, we can in all cases easily eliminate dp\ dp", dpi^ 

 from the corresponding differential equations and thus ol)tain a 



