EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 299 



that the surface abc is the surface D-E and therefore perpendicular 

 to Se, etc. Let tetr abed, trian abc, etc. denote the volume of the 

 tetrahedron or the area of the triangle specified, sin(ab, be), 

 sin (abc, dbc), sin (abc, ad), etc. the sines of the angles made by the 

 lines and surfaces specified, and [BCDE], [CDEA], etc. the volumes 

 of tetrahedra having edges equal to the tensions of the surfaces 

 between the masses specified. Then, since we may express the 

 volume of a tetrahedron either by ^ of the product of one side, an 

 edge leading to the opposite vertex, and the sine of the angle which 

 these make, or by f of the product of two sides divided by the 

 common edge and multiplied by the sine of the included angle, 



tetr bcde : tetr acde 



be sin (be, cde) : ac sin (ac, cde) 



sin (ba, ac) sin (be, cde) : sin (ab, be) sin (ac, cde) 



sin (ySe, PSe) sin (aSe, a/3) : sin (ySe, aSe) sin (/3(Se, a/8) 



tetr yPSe tetr paSe tetr ya Se tetr apSe 



trian pSe trian aSe ' trian aSe trian pSe 



tetr ypSe : tetr yaSe 



[BCDE]: [CD KA]. 

 Hence, 



V A : V E : v : v : : [BCDE] : [CDEA] : [DEAB] : [EABC], (641) 



and (640) may be written 



_ 



[BCDE] + [CDEA] + [DEAB] + [EABC] 



If the value of p E is less than this, when the tensions satisfy the critical 

 relation, the point where vertices of the masses A, B, C, D meet is 

 stable with respect to the formation of any mass of the nature of E. 

 But if the value of p E is greater, either the masses A, B, C, D cannot 

 meet at a point in equilibrium, or the equilibrium will be at least 

 practically unstable. 



When the tensions of the new surfaces are too small to satisfy the 

 critical relation with the other tensions, these surfaces will be convex 

 toward E ; when their tensions are too great for that relation, the 

 surfaces will be concave toward E. In the first case, TF V is negative, 

 and the equilibrium of the five masses A, B, C, D, E is stable, but the 

 equilibrium of the four masses A, B, C, D meeting at a point is 

 impossible or at least practically unstable. This is subject to the 

 limitation that when p E is sufficiently small the mass E which will 

 form will be so small that it may be neglected. This will only be 

 the case when p E is smaller in general considerably smaller than 

 the second member of (642). In the second case, the equilibrium 

 of the five masses A, B, C, D, E will be unstable, but the equilibrium 



