Problems in the Theory of Elasticity. 17 



and two similar equations obtained from this by cyclic per- 

 mutation of the variables. In these u, v, w are the rectan- 

 gular components of the displacement of the particle at the 

 point (.r, y, z) of the body, n the inward drawn normal, 

 6 the cubical dilation, and L, M, N the components of the 

 surface traction at the boundary point considered ; while 

 the constant k is given by 



fl being the velocity of propagation of longitudinal waves 

 and o> that of transverse waves in the body. The constants 

 X, fjL are those employed by Lame *, /j, being the rigidity 

 = pco 2 . In the equations involving the surface tractions we 

 have supposed the units so chosen that //, is equal to unity ; 

 and we shall throughout adhere to this assumption. 



If we denote the vector displacement of the point p of 

 the body by D or D(p), and the surface traction at the point s 

 by T or T(s), the equations for the equilibrium of the 

 particle at p are readily reduced to the single equation 



V 2 D + *graddivD = (1) 



or its equivalent 



a + l)graddivD-curlcurlD = 0; . . . (1') 

 while for the surface-point s 



— T(*)=2-^D+(A--l)ndivD + nxcurlD, . . (2) 

 an 



n being the unit vector in the direction of the inward 

 normal. It will be convenient to introduce the symbol P or 

 F(s) to denote the surface traction with its sign changed, i.e. 



»(f) = -T(,), 



whose value is given by the second member of (2). For 

 consistency of notation we shall throughout use the letters 

 p, q to denote points within the region considered but not 

 on the boundary ; while t, s; S, a will represent boundary 

 points, and dt, ds, &c. the corresponding elements of the 

 boundary. The surface 2 bounding the body separates the 

 finite inner region S from the infinite outer region S'. We 

 shall assume that 2 possesses everywhere a definite tangent 

 plane and two definite principal radii of curvature. Singu- 

 larities such as points and edges are excluded. 



* rt Le9ons sur la th^orie math, de l'6lasticit6 des corps solides." 

 Deuxieme Lecon. Paris (1852). 



Phil. Mag. S. 6. Vol. 32. No. 187. July 1916. C 



