EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



when i> D is sufficiently small the figure abed may be regarded as 

 rectilinear, and that its angles are entirely determined by its 

 tensions. Hence the ratios of t> A , i> B , v , V D , for sufficiently small 

 values of VD, are determined by the tensions alone, and for con- 

 venience in calculating these ratios, we may suppose p A , p E , p c to 

 be equal, which will make the figure abed absolutely rectilinear, 

 and make > D equal to the other pressures, since it is supposed that 

 this quantity has the value necessary for equilibrium. We may 

 obtain a simple expression for the ratios of v, i> B , v 0t V D in terms 

 of the tensions in the following manner. We shall write [DBC], 

 [DCA], etc., to denote the areas of triangles having sides equal to 

 the tensions of the surfaces between the masses specified. 



v : V B : triangle bdc : triangle adc 

 : be sin bed : ac sin acd 

 : sin bac sin bed : sin abc sin acd 

 : sin y8/3 sin Sa/3 : sin y8a sin 

 : sin yS@ 8/3 : sin ySa So. 

 : triangle yS/3 : triangle ySa 

 : [DBC] : [DCA]. 



a 



Hence, 

 where 



v : v :: [DEC] : [DCA] : [DAB] : [ABC], (634) 



may be written for [ABC], and analogous expressions for the, other 

 symbols, the sign ^/ denoting the positive root of the necessarily 

 positive expression which follows. This proportion will hold true 

 in any case of equilibrium, when the tensions satisfy the condition 

 mentioned and v^ is sufficiently small. Now if PAPEPC> PD 

 will have the same value, and we shall have by (627) TT V = 0, and 

 by (633) TFg = 0. But when V D is very small, the value of W s is 

 entirely determined by the tensions and V D . Therefore, whenever 

 the tensions satisfy the condition supposed, and V-Q is very small 

 (whether p A , p E , p c are equal or unequal), 



= W g = F v =^ D ^D -PA^A -Psv* -p G v G , (635) 



which with (634) gives 



+ [DAB] Po 



[DBC] + [DCA] + [DAB] 



(636) 



Since this is the only value of > D for which equilibrium is possible 

 when the tensions satisfy the condition supposed and v^ is small, 

 it follows that when > D has a less value, the line where the fluids 



