G2 Mr Macdonald, On the torsional [May 1, 



Monday, May 1, 1893. 



Peof. McK. Hughes, President, in the Chair, 



The following Communications were made to the Society : 



(1) On the torsional strength of a hollow shaft. By H. M. 

 Macdonald, M.A., Clare College. 



Saint- Venant in the eleventh chapter of his Memoir on the 

 Torsion of Prisms, 1855, deals with the case of hollow prisms, 

 but in the cases there considered the bounding surfaces are con- 

 centric. 



In the Phil. Mag., Jan. 1892, Mr Larmor investigated the 

 shear at the surface of a hollow in a prism, the dimensions of 

 the hollow being supposed small compared with its distance from 

 the outer surface of the prism. 



The object of the following is to consider the case of a prism 

 whose bounding surfaces are any two cylindrical surfaces of 

 circular section. 



1. Take as axes of x and y the radical axis and the line of 

 centres of the two circles in which a plane perpendicular to 

 the axis of the prism cuts it, and as axis of z the straight line 

 through their point of intersection parallel to the axis of the 

 piism. 



Then the displacements are 



u = — ryz') 



v= Tzx\ (1), 



10= (^J 



where T is the twist per unit length of the prism and <^ satisfies 

 the equation 



^ + ^ = 0.. (2) 



dx'^ dy'^ ^ ' 



throughout the matter of the prism, and 



^If +^'^^ = ^(^2/-'»-^) ■.■•-(3) 



at every point of the boundary of a cross-section, /, m being 

 the direction-cosines of the outward-drawn normal. 



Let -y^r be chosen so that ^ and i/r are conjugate functions of 

 ^ and 2/, then ^ satisfies the equation 



^^^-^ (2-) 



