508 



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



From equation (11) we see that — depends on 



or, putting N equal to f E, this becomes 



5^-3 



a 1 



5-^ + 13 



Hence, remembering as before that 6 2 has only values between and 

 5" 4a' 2 , we see that the last fraction varies between —0*23 and -f0"47. 

 Hence, taking b small, which gives the greatest values for and 



gives a value for ^ not less than half as great as if we had taken 



the largest possible value of h. 



a 



Finally, therefore, if b varies from to —p= — , the best practical 



Vjl — sin a 



value of b is an extremely small one, or the strip should be wound as 

 in figure 3. 



a 



If b is greater than -7 — — , conditions other than those given 

 v 1 — sin a. 



above may make 0, — , and — have their maximum values ; but, since 

 f d 



the difficulty of manufacturing metal springs of the form shown in 

 fig. 4 must necessarily render their employment but very limited, a 

 mathematical examination of this problem has not much practical 

 value. We therefore merely mention that in this case calculation shows 

 that 



<t =V a tan . (*±» - ± ) * aA ™* — * . . (12), 



where / is the maximum stress anywhere in the section, the maximum 

 stress in this case not occurring at the extremity of the b diameter. 



The general conclusions therefore arrived at are, that in order, with 

 a given axial force to obtain a large amount of turning of the free end 

 of the spring, combined with small maximum total stress in the 

 material, and not too much axial motion of the free end of the spring, 

 the strip of elliptic section should be as long and as thin as possible, 

 should be wound in a spiral such that the osculating plane makes an. 

 angle of 40° to 45° with a plane perpendicular to the axis of the spiral, 

 and so that the smaller diameter of the elliptic section is at right 

 angles to the axis of the spiral. 



