Manchester Memoirs, Vol. xlvi. (1902), No. 15. 1 1 



and within this surface suppose that 

 u =0, 



^^='rX (17) 



the conditions will be satisfied at points within the cylinder 

 as well as outside it, and the displacements will be con- 

 tinuous at its surface. 



At this surface, of which the section is an ellipse 

 having its minor axis horizontal, and the eccentricity- 

 equal to sechy, the stresses will be discontinuous, the 

 differences of the stresses being given by 

 S ^ - 2orjutanhysinOcosy, 



U= - 2ff/jCOS-6, 



where Jr = tsinh7cos0 ; ^■ = tcosh'ysinf^. 



As the elliptic curve of discontinuity travels parallel 

 to the surface at the rate q, we shall have a quantity of 

 matter crossing an arc ds of this curve at the rate i>qdz, 

 and receiving an instantaneous change of velocity 

 measured by — 2(t$'cos'^6) as it crosses. The rate of change 

 of momentum is therefore - 2gnq''co?r^dz for the element 

 ds in the direction of the axis of j. 



The differential stress at the surface of discontinuity 

 at the same element is in the same direction, viz. — 



{/Z7+ nS)ds 

 where (/, 0, n) are the direction cosines of the normal 

 measured inwards, and therefore 



Ids = dz and nds = - dz. 



Hence 



(JV+iiS)ds 

 = - 2(T/u(cos20 + \.a.x\\\~y?,m-d)dz, 

 = - 2a^(sech2ycos-(^ + tanh'y)^s, 

 = - 2a{p^-cosW + ii\.anh'-/)dz, 

 since pg" =/zsech^7. 



