344 



MK. L. N. G. FILON OX THE RESISTANCE To 



<|uantity E I projxise to cull the "efficiency" or "usefulness" of the shaft. It is 

 equal to (S /M )/(S/M), and gives us the ratio of maximum stress to tnrsional rigidity. 

 compared with the same ratio for the circle of equal area, that is, it gives us a 

 measure of how much torsion we may put into the shaft without impairing its 

 elasticity. Thus if M be the maximum torsion moment which the given shaft will 

 bear without failure of elasticity, M the maximum torsion moment which the shaft 

 of equal circular section (made of the same material) will stand, then the corre- 

 sponding stresses, S, S are each equal to the limiting elastic stress of the material 

 and 



M/MO = efficiency = E, 



or the limiting torsion moment of a shaft of any section is obtained from that of the 

 circular section of the same area by merely multiplying by the "efficiency" as thus 

 defined. 



TABLE of " Efficiency " of the sixteen given sections. 



A glance at the above suffices to show that the efficiency of these sections is, in 

 general, about one-half, that is, on the whole, this form of shaft is about half as useful 

 as the shaft of circular section. The zero efficiency in the case of the slit (ft = Tr/2) 

 is due to the infinite stress in the keyway. 



20. Analysis for the Sections bounded by one Elliptic and one Hyperbolic Arc. 



I now pass on to the consideration of cross-sections, bounded by an ellipse and a 

 single branch of a confocal hyperbola. Such sections are shown in figs. G-8. 

 In this case we shall find it more convenient to define and 17 by the equations 



x = c cosh cos - 

 y = c sinh f sin 77 



(23) 



where 



, < rj < /3. 



