342 



MK. L. N. G. FILON OX TIIK RESISTANCE TO 



Hence we have proved that the only fail-points are those which we have already 

 investigated. 



18. Deductions from the above and Criticism of BOUSSINESQ'S Proof of the 



Position of Fail-Points. 



Thus we see the study of these symmetrical sections is extremely instructive as 

 regards the position of the fail-points. They show us the connection between the 

 rectangular section and the section with a neck, and they give us the limiting cases, 

 when the four fail-points coalesce into two, and vice versd. The four fail-points 

 begin to occur after the ratio of the long to the short axis of the section exceeds a 

 certain value, which depends upon the angle of the bounding hyperbolas. As the 

 indented appearance of the section becomes more obvious, that is, as ft increases, this 

 limiting value becomes greater and greater until, when the angle between the 

 asymptotes is less than 73 the fail-point is always at the vertex of the hyperbolas. 

 But in no case are the fail-points on the convex sides of the sections, unless the ellipses 

 are so flat that the points A are nearer to the centre than the points B. 



M. BOOSSINESQ has given (' Journal de Mathematiques,' SeYie II., vol. 16, p. 200) 

 a sketch of a proof that the fail-points must be sought for " sur les petits diametres 



Fig. 11. 



des sections." A's this statement is in opposition with the results of the present 

 paper and with DE SAINT- VEN ANT'S results for the rail sections already mentioned, I 

 venture to suggest that M. BOUSSINESQ'S reasoning hardly holds in the case of 

 sections part of whose contour is convex and part concave, for the following reason. 

 The problem of torsion is mathematically analogous to that of a cylindrical vortex of 

 uniform strength, whose cross-section is that of the shaft considered. The motion 

 l)eing in two dimensions we have a stream function t/, and the resultant stress at any 

 point in the torsion problem is the same as d\jj/dn in the hydrodynamical analogy, dn 

 leing an element of the normal to any stream -line. 



Now if we draw the stream-lines for equidistant values of t/, they will, says 

 M. BOUSSINESQ, "reproduce the irregularities of the contour, but more and more 

 faintly, so that the curves are spaced at greater intervals along the large, then along 

 the small diameters." In consequence, d\jt/dn, or the stress, is greater in the latter 

 case. The above argument assumes that the same number of stream-lines cross each 



