to Torsion of certain forms of Shafting. 



481 



The changes in the stresses are shown by curves accompanying the 

 paper, in which the abscissa represents the quantity a, whose hyperbolic 

 cosine and sine are proportional to the major and minor axes of the ellipse 

 respectively, and in which the ordinates represent the stresses at A, B, F, 

 divided by the maximum stress of the circular section of equal area. 

 The curves are, in certain parts, only roughly drawn, but they suffice to 

 show the manner in which the stresses vary. It is seen that the stress 

 at«B separates from the maximum stress after the critical value a = 

 1*225, and gradually diminishes, compared with stresses at A and F. 



This result might have been expected from the investigations of de 

 Saint- Venant upon certain sections bounded by curves of the fourth 

 degree. 



These investigations appear, however, not to have been sufficiently 

 noticed. Thomson and Tait in their ' Natural Philosophy,' and 

 Boussinesq in his researches on Torsion,* both conclude that the fail- 

 points are at the points of the cross-section nearest to the centre, and 

 Boussinesq even gives an apparently general proof of this proposition. 

 His proof, however, is subject to certain restrictions, which I point out, 

 and which prevent it from being applied to the sections I am dealing 

 with. 



The sections are sensibly less useful than the circular section, their 

 torsional rigidity being always diminished and the maximum stress 

 very often increased. This remark, I may add, applies to all the 

 sections dealt with in this paper. 



This usefulness or efficiency decreases as the neck of the section 

 becomes more narrow, as indeed might have been anticipated. 



Other sections worked out are those corresponding to angles between 

 the asymptotes of 90°, 60°, and 0° ; in the latter case the sections 

 degenerate into ordinary elliptic sections, with two straight slits or 

 indefinitely thin keyways, cut into them along the major axis, as far as 

 the foci. The stress at the foci, however, is then theoretically infinite. 



It is interesting to see how, as we make the bend round the foci 

 sharper, the values of a, for which the two fail-points break up into 

 four, become larger and larger, until, when the angle between the 

 asymptotes of the hyperbolas is less than 73°, the greatest stress always 

 occurs at the neck of the section. 



The limiting case of such sections, when the angle between the 

 asymptotes is very small, and the eccentricity of the ellipse nearly unity, 

 the distance between the foci being very great, gives us the rectangle. 



I then pass on to the section bounded by one ellipse and one confocal 

 hyperbola. In the limiting case, when the foci coincide, we obtain the 

 sector of a circle. Of this I have worked out numerically three cases, 

 in each case taking two ellipses. 



1. The semi-ellipse. 



* ' Journal de Matliematique3,' serie 2, vol 16, p. 200. 

 VOL. LXV. 2 K 



