470 RICE 



AHT. K 



must conclude that the three factors in the integrand multiply- 

 ing 8x, 8y, 8z are severally zero, and so we arrive at [378]. (The 

 doubly accented direction-cosine symbols used in the argument 

 for the sake of distinction between s' and s" are, of course, not 

 required any longer.) The expression referred to in [379], and 

 the two similar expressions are of course the expressions in 

 (29a) of this article, except that the former are the com- 

 ponents of the stress-action at a surface on an area which was 

 unit size in the state of reference, the latter on one which is unit 

 size in the state of strain. The interpretation then put on [378] 

 is obviously necessary for the equilibrium of an internal thin 

 layer of the solid, bounded by two surfaces parallel and near to 

 the surface of discontinuity, one in one part of the solid and one 

 in the other. 



8. Commentary on Pages 191-197. Discussion of the Four 

 Equations of Equilibrium. Let us now resume the commen- 

 tary on details in pages 191-197. The equations [377] are a 

 particular case of (29) of this article in which the compo- 

 nents Fx, Fy of the force per unit volume are zero and Fz = —gV. 

 (Remember that OZ is directed upwards so that gravity is in 

 the negative direction of OZ.) The meaning of the remarks 

 which immediately follow concerning [375] and [376] may 

 perhaps not be obvious to all readers at first sight. When we 

 proved these equations in this exposition, we assumed that the 

 solid was in equilibrium, but strictly this assumption was un- 

 necessary. For if we refer once more to the proof leading to 

 equation (30) and do not assume equilibrium, we must put the 

 couple on the element of volume arising from the stresses of the 

 surrounding matter and from the body forces on it equal, not to 

 zero, but to the sum of the moments of the mass-acceleration 

 products of the various particles of the element; i.e., to the 

 product of the moment of inertia of the element and the angular 

 acceleration. Now, without going into too much detail, this 

 moment-sum, like the moment of the body forces, involves terms 

 which have as a factor the product ^rjf and a length of the same 

 order of magnitude as ^, 77 or f . In consequence it is evanescent, 

 just as is the moment of the body forces, in comparison with the 

 moment of the stress-actions, and the same result follows as 



