114 



Mil. I-. N. C. Fll.oN "N TIIK KKSISTANVK To 



Of this I have worked out numerically three ca,ses, in each case taking two ellipses 



(1.) The semi-ellipse (fig. 6). 



(2.) The ellipse with a keyway cut into it of the shape of a rectangular eonfocal 

 hyperbola (tig. 7). 



(3.) The ellipse with a single slit cut into it (fig. 8). 



The most striking of the results is in reference to the reduction of the torsional 

 rigidity of the ellipse in case (3). This reduction of rigidity decreases rapidly as the 

 depth of the notch decreases. 



The rigidity, which is reduced hy as much as 23 per cent, when the depth of the 

 keyway is as great as '6 (semi-major axis), falls only about 1 per cent, when this depth 

 is '12 (semi-major axis). 



Possibly this may throw some light on the fact that the effect of cutting such slits 

 into the material does not always give in practice the reduction in the torsional 

 rigidity which should have been expected from SAINT- TENANT'S results for the circle. 

 Clearly the depth of the keyway is a factor of the very first importance, and key ways 

 of moderate depth will produce a comparatively small effect on the torsional rigidity. 



It is also shown that the effect of cutting two equal and opposite slits is practically 

 equal, in the two cases which I have calculated (namely, a = 7r/6 and = 7r/2), to twice 

 the effect of a single slit. 



It seems, therefore, that the study of these sections brings to light several 

 interesting facts in the theory of elasticity, and Avill well repay the trouble involved 

 in dealing with the long and somewhat tedious algebra and arithmetic which lead to 

 these results. 



3. Statement of Notation, &c. 



In what follows the axis of z will be taken parallel to the generators, the axes of x 

 and y in one of the terminal cross-sections. 



The origin, however, will not necessarily be at the centroid of the cross-section. 



The shifts of any point of the material parallel to the axis will be denoted as usual 

 by u, v, w, and for the stresses I shall use the notation of TODHUNTEH and PEARSON - 



' History of Elasticity,' rs denoting the stress, parallel to r, across a plane element 

 perpendicular to s. 



Then if, following SAINT- VENAXT, we suppose the terminal cross-section z = to 

 be fixed (that is to say, = 0, v = 0, but w = 0), then if T be the angle of torsion 



per unit length, 



v = rxz, u TIJ~. ( I ). 



Iff* be the modulus of rigidity, 



yz '/" 



= + TX, 

 p, ,/// 



ra 

 /* 



the 



(2), 



and all the other stresses are zero. 



