522 Dr. C. Chree. On the Determination of [Dec. 21, 



(13).* 



Confining ourselves to the case where the material is symmetrical, 

 about (Xz, we have 



?73i = ??32 = -n, say, 7712 = 7721 = V"l xj.v 



^w = i?28 = i?", say, E 2 = EI = E' J " 

 Also, writing E for E 3 it may be shown that 



WE' = if/E ........................... (15). 



Suppose, now, the solid to be a hollow right prism exposed to 

 uniform tension P over its flat ends, S, and to uniform pressures, pi 

 over its inner surface Si, and p e over its outer surface S e , then 



where A, p, v are the direction cosines of the normal drawn from the 

 solid, and the suffix denotes the surface. 



Taking the origin in one end of the prism, and supposing its length 

 Z, we have z = over the one flat end and z = I over the other ; thus 

 S denoting the area of one end, 



PadS = PIS = Pv, 



where v is the volume of the material. 

 Over the two prismatic surfaces 



where wt and -sr e represent the perpendiculars from some internal 

 fixed point on the tangent planes to the two surfaces. But obviously 



where vt and v e are the volumes included within S t - and S e respectively. 

 Thus (16) gives at once 



E?;<7 = Pv + 2rj (p e v e - jpii). 



Clearly g = 8l/l, where 81 represents the mean change in length of 

 all the longitudinal " fibres " of which the prism is composed. Thus 



As in the isotropic circular cylinder, there are three principal cases : 

 (i) p e = 0, Pv =p& (18), 



* ' Camb. Phil. Soc. Trans.,' vol. 15, Eqn. (15), p. 317. 







