472 RICE ART. K 



fluid (ni being the potential of the sohd substance in the Hquid) ; 

 €v,r]v and r, to the sohd. The subsequent discussion is Umited to 

 the case of a sohd body which is not only homogeneous in 

 nature, but also homogeneous in its state of strain. The first 

 point considered by Gibbs is concerned with the conditions 

 under which this latter proviso is compatible with a uniform 

 normal pressure over any finite portion of the surface. (The 

 effect of gravity, the only body force considered in the general 

 discussion preceding, is disregarded as negligible in producing 

 heterogeneity of strain or variation in the value of pressure at 

 different points of the surface.) This leads at once to Gibbs' 

 discussion concerning the three principal axes of stress on pages 

 194 and 195. We need not comment on this, as we have already 

 proved the necessary propositions in our exposition, starting 

 from an expression similar to [389]. Gibbs' proof is an analyti- 

 cal one based on the methods of the calculus as applied to 

 questions of maximum-minimum values of functions of several 

 variables, and will be easily followed by those acquainted with 

 these methods, whereas the method we have used, being 

 based on the elementary geometrical properties of the stress- 

 quadric will probably be intuitively perceived by those not so 

 well versed in mathematical analysis. Actually, if we revert for a 

 moment to the form of equations [382] which we have written 

 above, the conclusions arrived at in the paragraph which 

 includes the equations [393], [394], [395] can be obtained in a 

 very direct and suggestive manner. Equations [382] in our 

 form can be written thus : 



(Xx' + Anp)a' 4- (Xr> + Ay,p)l3'^ 



+ (Xz' + A,sp)Y = 0, 



(Yx' + Anp)a + {Yy> + A,,p)^' 



+ (Yz' + A2zp)y' = 0, 



{Zx' + A3ip)a' -t- {Zy + A32PW 



-f (Z^, + Anp)y' = 0. 

 If the solid is in a given homogeneous state of strain, Xx', . ■ ■ Zz', 



> [382a] 



