474 RICE 



ART. K 



that once more only a pair of orientations, given by a^, fi-i, 

 72'; —OC2, —^2^ — ji' qm6. oii , ^2! , 73'; —0:3', — jSa', —73', are com- 

 patible with these pressures respectively and the given state of 

 strain. Furthermore, it can be proved from the equations that 



«!'«/ + iS/iSa' + 7/72' = 0, 

 cii'az' + /32'/33' + 72'73' = 0, 

 az'ai' + /33'/3i' + 73'7i' = 0, 



showing that the three directions are normal to each other; but 

 the proof would lead us too far into the theory of such deter- 

 minantal equations. Indeed, as doubtless many readers know, 

 the analysis is quite similar to that employed in analytical 

 geometry when determining the directions of the three principal 

 axes of a quadric surface, and in fact Gibbs derives the result 

 by a direct appeal to the existence of the three principal axes of 

 stress which will, of course, have the same directions at all points 

 of the solid if the strain is homogeneous. These directions 

 are in fact the directions on', fii , 71'; 0:2', ^2, 72' and az, 183', 73'; 

 and pi, P2, Ps are respectively —Xx, —Yy, —1z if the analysis 

 of the stress-constituents has been referred to these principal 

 axes as the axes of reference in the state of strain. (Xy, Y z, Zx, 

 etc. are of course each zero in such case. In order to avoid con- 

 fusion we have thus far had to use suffixed symbols for the 

 three pressures instead of accented symbols; for the use of ac- 

 cented symbols to indicate measurements in the state of refer- 

 ence makes it awkward to use them for any other purpose, such 

 as distinguishing three different values of a quantity. How- 

 ever, as the subsequent treatment will not require the use of 

 direction-cosine symbols, we shall revert to Gibbs' notation 

 and substitute p', -p", jp'" for pi, p-i, pa.) 



In this way the important conclusion emerges that only three 

 fluid pressures are compatible with an assigned homogeneous 

 state of strain of the solid in contact with the fluid, and if one of 

 these pressures is established in the fluid, the solid, if equilib- 

 rium is to be preserved, can only be in contact with it at a pair of 

 plane surfaces whose normals are opposite to one another in 

 direction. Of course, this is a general statement; there are 



