6 = 



du 

 '"dr 1 



u 



J r 



ldv 

 + rd0 l 



dw 

 9 ~dz" i 







| 



a- 



dv 

 Z di + 



1 dw 

 rd0 ; 



, dw 

 dr 



, du 



d v - 



r 

 dr 



1 du 

 + rdd' 



1 



82 Mr. Co Chree on Bars and 



In considering the strains, stresses, and surface-conditions 

 we may employ Thomson and Tait's notation and results, 

 looking upon the radius vector through any point as the axis 

 of x at that point. Thus the strains are 



(8) 



These present no difficulty if it be noticed that the axes of a, 

 as above defined, at adjacent points are inclined at an angle dO. 

 The corresponding stresses are 



P = (m — n)8+2ne; Q = (m — n)8 + 2nf; B,= (m — n)8 + 2ng; 

 S = na; T = nb; JJ = nc (9) 



For the surface-conditions, denoting by X, //,, v the cosines 

 of the inclination of the normal at the point to the radius 

 vector, to the perpendicular to the radius vector in the cross 

 section, and to the axis of z respectively, we have* 



\P + /-tU + vT= component of surface-force along 

 radius vector, 



XU + /u<Q + vS = component perpendicular to above [> . (10) 

 in cross section, 



XT + /jlS + vR= component parallel to axis of z. 



For a right circular cylinder, whose axis is axis of z, we have 

 on the curved surface, 



fji = = v, X=l ; 

 on the flat end, 



X = = /tt, v=l. 



Thus for a right circular cylinder exposed to no surface-forces 

 we must have 



P = U = T = on the curved surface, 1 n *. 



T = S = R = on the flat ends. . J ' Ui) 



Let us consider first the case of a right circular isotropic 

 cylinder, whose axis is taken as axis of z, fixed at the end z = 0, 

 and exposed to tensions or pressures on its other end symme- 

 trically distributed about the axis. Then obviously v = Q 

 throughout, and u and W are independent of ; thus (2) is 

 identically satisfied, and (1) and (3) much simplified. Sup- 

 posing these terminal forces independent of the time, the 



* See Thomson and Tait's ' Natuial Philosophy/ §§ 662 and 734. 



