306 Profs. W. E. Ayrton and J. Perry. [Feb. 14, 



so that the greatest stress can never occur at the end of the semi- 

 diameter a, but may be very near to it. 



On the supposition then that, relatively to a, b has any value from 



a 



to / ; — , the greatest stress in an elliptic section occurs at the 

 v 1 — sin a. 



end of the semi- diameter b, which is parallel to the axis of and this 

 is the case we will first confine ourselves to. 



Maximum 'Rotation in Relation to Permanent Set. — If, now, /is the 

 greatest stress at an'y point of the section, 



_ x I sin a. cos a fa? + b 2 



f z b 1+sin* V N« 2 



£)••••.■ < 10 >- 



The conditions that make this a maximum are those which for 

 a given axial force applied to our spring produce the greatest 

 amount of turning of the free end with the least amount of stress on 

 the material, and therefore with the least chance of permanent set. 

 And as regards the value of «, it is clear that 



sin«= — \ + \^ 5, 



or a =38° 10', 



will give the greatest value. 



Maximum Rotation compared with Axial Motion. — From equations 

 (3) and (4) we have 



fl3+z> 3 _ 4 



_tan a, Na 2 E 



1 r . { 



- — + — • tan- a. 



Na2 E 



and the conditions that make this a maximum are those which for a 

 given axial force applied to the spring give the greatest amount of 

 turning of the free end of it with the least amount of axial length- 

 ening. As regards the value of a., it can easily be shown that ~ will 



a 



be a maximum when 



tan a.— y //E 



N V 



If b is small in comparison with a, which is a condition, as already 

 explained, we are led by facility of construction to adopt, then 



tan«=J 



As a rough approximation, taking N the modulus of rigidity at two- 



