to Torsion of certain forms of Shafting. 



429 



Clebsch has used such elliptic co-ordinates to solve the torsion problem 

 for hollow cylinders bounded by confocal ellipses, and de Saint- Venant 

 has applied conjugate functions to the same problem for shafts whose 

 sections are sectors of circles ; curvilinear co-ordinates have also been 

 employed by Mr. H. M. Macdonald,* but I am not aware that the 

 actual solution has yet been obtained for sections bounded by both 

 ellipses and hyperbolas. 



The work proceeds on lines analogous to those developed by Saint- 

 Venant himself in his solution of the problem of torsion for the cylinder 

 of rectangular cross-section. The strains and stresses are expressible 

 in terms of infinite series involving circular and hyperbolic functions. 



The boundaries of the section are given by constant values of £ and 

 7]. The values of'£ are taken to be ± a. 



The conditions from which the unknown quantity w (the shift par- 

 allel to the axis) is determined are 



d 2 w/dx 2 + d 2 w/dy 2 = 



throughout the sections ; and 



diu/dn + (mx - ly)r = 



along the boundary, where dn = an element of the outwards normal to 

 the boundary, r is the angle of torsion per unit length, and I, m are the 

 direction-cosines of dn. 

 Now in the present case 



dn = ±d£x(c Jj) 



where J = 3^5, at the boundary where £ - const, and 



dn = ±drjx (c J J) 



at the boundary where iq = const., the sign being determined so that 

 dn is positive. 



By adding suitable terms to w, we can reduce one or other of the 

 boundary conditions to the form 



dw\jdn = 



where w = W\ + suitable terms. 

 Suppose we make 



dwjdg = ; J = ±a. 

 Expanding now Wi in the form of a series, 



"A„ sin h f + K ) } sin "l+M, 



* ''On the Torsional Strength of a Hollow Shaft," ' Proc. Camb. Phil. Soc. ' 

 vol. 8, 1893, pp. 62 et seq. 



