380 



MR. L. N. G. FILON ON THE RESISTANCE TO 



/0sinh2 + sin2/8\ 

 TABLE of 



\ 7T / 



whence we obtain the following table giving us M/M : 



When we look at this table, we observe immediately that the torsional rigidity 

 decreases, compared with the torsional rigidity of the circular section, as we increase 

 a, that is to say, as we decrease the thickness of the neck with regard to the other 

 linear dimension. This indeed might have been expected, for it is clear that such a 

 process must weaken the rigidity enormously, inasmuch as it tends to render the two 

 halves of the section independent of each other. 



When a = 7T/6, ft = Tr/6, the ratio M/M is greatest. In this case the section does 

 not deviate very much from a square. (For very small values of a and ft the section 

 is, of course, a rectangle.) This result shows us therefore that, so far as torsional 

 rigidity is concerned, the square is a more efficient form of section than any one of 

 those dealt with in the present paragraph. 



That the circle is a more efficient type of section for rigidity is quite evident from 

 the table, since all the values in it are less than unity. 



It may be interesting to note what are the values of M/M for the full ellipse. 

 When we use the values given by SAINT- VENANT in his memoir on torsion 



M 



= and M. = 



M, 



= ~TT~is = tanh 2a, if b/a = tanh . 





