466 J. W. Gibbs — Equilibrium of Heterogeneous Substances. 



TFv^==0, when the critical relation is satisfied and v^ small. This 

 gives 



Pk= • (640) 



In calculating the ratios of i'a, "^b, ^c-, "d, ^e, we may suppose all the 

 surfaces to be plane. Then E will have the form of a tetrahedron, 

 the vertices of which may be called a, b, c, d, (each vertex being 

 named after the mass which is not found there,) and u^^ ^b? ^c^ ^d will 

 be the volumes of the tetrahedra into which it may be divided 

 by planes passing through its edges and an interior point e. The 

 volumes of these tetrahedra are proportional to those of the five 

 tetrahedra of the figure a fi y d e, as will easily appear if we recollect 

 that the line a b is common to the surfaces C-D, D-E, E-C, and there- 

 fore perpendicular to the surface common to the lines y d, 6 e., e y^ 

 i. e., to the surface y S a, and so in other cases, (it will be observed 

 that y, 6, and e are the letters which do not correspond to a or b) ; 

 also that the surface a b c is the surface D-E and therefore perpendic- 

 ular to S E, etc. Let tetr abed, trian abc, etc. denote the volume of 

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

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

 lines and surfaces specified, and [B CD E], [C D E A], etc., the vol- 

 umes 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 mixltiplied by the sine of the included angle, 



Vf^-.v^:: 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 {yd£, /3d e) sin {ade, ap) : sin (yds, ade) sin {pSs, afi) 

 tetr yfidE tetr (dade tetr yade tetr afide 

 trian (ids trian ade ' trian ade trian fids 



: : tetr y(3de : tetr yade 



::[BCDE]:[CDEA]. 



Hence, 



?JA:«'B:'yc:^D::[BCDE]:[CDEA]:[DEAB]:EABC],(64l) 



and (640) may be written 



_ [B C D E1pa+ [CDE A]79b -t- [D E a B] />c+ [E Aj^Cj^p .g^g) 

 ^^ [BCDE]-h[CDEAJ + [DEAB]-f [EABC] 



