94 Prof. Tait on the Importance of 



As dp may have absolutely any direction, this is equivalent to 



where c// is the conjugate of c/>. 



II. The fundamental kinematical theorem is easily obtained 

 from the consideration of the continuous displacement of 

 the points of a fluid mass. (It is implied in the word " con- 

 tinuous " that there is neither rupture nor finite sliding.) 



If cr be the displacement of the point originally at p, that 

 of p + dp is 



a + d(7 = (T — ^dp^ . cr ; 



and thus the strain, in the immediate neighbourhood of the 

 point p, is such as to convert dp into 



dp — Sdp \7 . cr. 



Thus the strain-function is 



If this correspond to a linear dilatation e, and a vector-rota- 

 tion e, both being quantities whose squares are negligible, we 

 must have 



yjrr — (1 + e)r 4- Ver. 

 Comparing, we have 



— St\7 . a- = er + Yer, 



from which at once (by taking the sum for any three direc- 

 tions at right angles to one another), 



Vo-=-2(V)+2e; 

 so that 



Sv°" represents the compression, 

 and 



^V^cr „ „ vector-rotation, 



of the element surrounding p. 



By the help of these expressions we easily obtain the stress- 

 function for a homogeneous isotropic solid, in terms of the 

 displacement of each point, in the form 



<pco= — n(Scoy.cr + ySft)cr) — (c — fn)o)Svcr ; 



where n and c are, respectively, the rigidity and the resistance 

 to compression ; and <pa> is the stress, per unit of surface, on 

 a plane whose unit normal is a>. 



III. The fundamental physical relation is that expressing 

 conservation of matter, commonly called the equation of con- 

 tinuity. We have only to express symbolically that the 

 increase of mass in a finite simply-connected space, due to a 

 displacement, is the excess of what enters over what leaves the 



