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Mr. L. N. G. Filon. On the Resistance 



the differential equation and the first boundary condition are identi- 

 cally satisfied. 



When this value is substituted in the second boundary condition, we 

 get an equation expressing a given function of £ in a series of sines of odd 

 multiples of 7r£/2a, between the limits -fa and - a. 



But such an expression can be definitely obtained by a method 

 analogous to that for Fourier's series. Comparing coefficients, we 

 obtain relations which determine completely all the constants in the 

 expression of Wi. 



w is then known. The shears and torsion moment are then deduced 

 by differentiation and a double integration. 



The cross-sections which are dealt with in the present paper, are of 

 very great generality, and they include as special cases many of the 

 cross-sections which Saint- Venant has worked out, for instance the 

 rectangle and the sector of a circle. 



The first section of which I treat is that bounded by an ellipse and 

 two confocal hyperbolas. Although the analysis is worked out for the 

 case where the two hyperbolic segments are not symmetrical, I have 

 not given any numerical examples of this case, as the sections obtained 

 by taking two hyperbolas curved the same way do not correspond to 

 any interesting^)ractical case : the section is too broad at the ends and 

 too narrow at the bend, to be any fair representation of the angle 

 iron. 



The section bounded by an ellipse and the two branches of a con- 

 focal hyperbola is, on the other hand, an approximate representation 

 of a well-known section, much used in engineering practice, the rail 

 section. This section I have worked out for various values of the 

 eccentricity of the ellipse, and of the angle between the asymptotes of 

 the hyperbola. The four sections where this angle is 120° give the 

 best representation of the rail section. 



The numerical results are tabulated so as to show the ratio of the 

 torsional rigidity of this section to that of the circular section of the 

 same area, and also the same ratio for the maximum stress. 



The ratio of these two ratios gives us a kind of measure of the use- 

 fulness or "efficiency" of the section. 



In the case of the latter class of sections, I have investigated at 

 length the position of the fail-points, or points of maximum strain and 

 stress, the maximum strain, in the case of torsion, being coincident 

 with the maximum stress. It is found that for the two smaller ellipses 

 the maximum stress occurs at the point B, where the section is 

 thinnest. For the two larger ellipses, the maximum stress occurs at 

 four points, F, F, F, F, symmetrically distributed round the contour, 

 and lying on the broad sides of the section. The critical section, 

 when these two cases pass into one another, can be calculated, and is 

 shown in the paper. 



