1893.] strength of a hollow shaft. 67 



than the greatest shear in the solid cylinder, so that the existence 

 of a small cylindrical cavity at a distance from the axis greater 

 than half the radius reduces the strength of the cylinder in the 

 ratio a : 26 approximately. 



4. When rj = ^ — ma, ^=0, s = t6u + i-j v, neglecting 

 powers of a, that is, at a distance from the surface of the cavity 

 lij] — iVa', the shear is t6 -jl + f- L This shews that the 



shear diminishes rapidly in the neighbourhood of the cavity ; 



5t6 

 e.g. a = 26, at dista^nce from surface of cavity a', s — -j- , at 



distance 3a' 5=^-^x6 etc., or a = — r , at distance ^ , 5 = -^r- , 

 15 2 z d 



5a' 97 



at distance — , s= -^ rb etc. Hence we may suppose any 



number of cavities to exist provided that their distance apart 

 is large compared with their radius, and the strength of the 

 cylinder will be determined by the cavity which is farthest 

 from the axis. The geometrical axis of the cylinder will form 

 a helix whose axis will be that of the centroids of the sections. 



From the expression for w we observe that the line of centres 

 of each section suffers no displacement parallel to the axis of z, 

 and that every other element of the section does, the displace- 

 ments of points symmetrical with regard to this line being of 

 opposite sign. 



5. If we take as bounding surfaces 7j = a and r} = — jS, and 

 the axis of ^ as axis of torsion, we obtain 



■xjr = 2tc^ coth a 2 (— )" e~"'' cos n0 from tj = a to rj — ^ , 

 1 



and t/t = + 2tc^ coth^ 2(— )"■ e+"'' cos n^ from t] = — /3 to r; = — on , 

 1 



or -v^ = Tcy coth a from rj =a to y—cc, 



and 1^ = — Tcy coth /3 from rj = — jB to ^ = — oo; 



hence in this case, which is that of a prism formed of two solid 

 circular cylinders outside one another, w = tcx coth a from 77 = a to 

 Tj = CO , and tu = — rex coth yS from 77 = — /3 to 77 = — go. 



The twisting couple is given by 



N TTT , , ,,\ 



n 1 



