332 MR. L. N. Q. FILON ON THE RESISTANCE TO 



The values of S A //*rc, as calculated from the formula (21) above, actually turn out 

 to be negative. The sign is, however, of no importance. 



Several important results are seen to follow at once from these tables. 



In the first place, when a keyway cut into a shaft of elliptic cross-section reduces 

 to a mere slit, the stress at the inner extremity of the keyway is seen to be infinite, 

 although this keyway is the limit of a single continuous curve and not of two curves 

 making a sharp angle, as is the case for a slit along a radius of a circle, obtained as 

 the limit of a keyway of the shape of a sector of the circle. 



Such slits are thus bound to produce rupture or plastic flow of the material at 

 their deepest points, whatever be the manner in which we approximate to them in 

 practice. 



The second point of importance, which these tables bring out clearly, is that the 

 maximum strain and stress do not always occur, as most of the results obtained by 

 DE SAINT- VENANT would lead one to suppose, and as THOMSON and TAIT (' Natural 

 Philosophy,' vol. 1, Part II., 710), and BOUSSINESQ ('Journal de Mathdmatiques,' 

 Se"rie II., vol. 16, p. 200) assumed, at the point of the boundary nearest the centre. 



Indeed SAINT- VENANT himself, in his edition of NAVIER'S ' Le9ons de Mdcanique ' 

 ( 33, p. 313), has given an example to the contrary, and it happens that the section 

 dealt with in this example is closely analogous to the sections of fig. 2 in this paper. 

 The shape of the section is reproduced in fig. 11 (p. 342), from SAINT- VENANT'S ' Le9ons 

 de NAVIER.' He calls it a " section en double spatule analogue d celle d'un rail de 

 chemin de fer." He gives two numerical examples in which the ratio of breadth to 

 length of the section is '20 and '14, corresponding for our sections, when ft = Tr/6, to 

 a. = l'G47 and a = 1'985 respectively. He finds in these two cases that the fail- 

 points are not at the point of symmetry on the contour which is closest to the centre, 

 but at points on the contour at a distance from the axis of symmetry of '46 and '52 

 of the half-length respectively. 



Now it is easy to see from the tables above that the result which one would expect 

 according to the ordinary rule, namely S B > S A , does hold in fifteen out of the sixteen 

 cases, but there is one exception in the case of the section a = 27T/3, ft = 7r/6, when 

 the greater of the two stresses is found to occur at A, the point further from the 

 centre. 



I was much struck at first by this apparently solitary deviation from the rule, and 

 was inclined to ascribe it to some error in the arithmetic. 



In order to test this, I calculated the values of S A and S B for the neighbouring 

 section, ft = Tr/6, a. = 37T/4. I found 



= 2-4333, - S - = T4144, 



- fj,rc 



confirming the previous exception. 



I then took the expressions (21) and (22) for S A and S B , and tried to determine the 

 limits to which they tended, when a was made very great. 



