1884.] New Spring for Electric and other Instruments. 305 



We have now to consider for what value of y this expression is 

 a maximum. The possible values of y are between and b ; and 



b 2 



we see that if 1 — — cos 2 « is positive, / will be greatest when y has its 



a z 



greatest possible value, namely b. For sections 1, 2, 3, 4, therefore, 

 •of the strip shown in fig. 5, where AB is the axis of the spring, 



the greatest stress will in each case be at B. If, however, b be very 

 large compared with a, which is the case in the section 5, fig. 5, then / 

 will not have its greatest value at B. 

 We find also that when 



a< b cos «, 



/ has its greatest value when 



_ ab 2 sin a. 



V ~ cos 3 *-a*)QP-a*) (9 '* 

 Now this value of y is greater than b so long as 



a 2 b 2 sin^>(& 2 cos 2 «-a 2 )(& 2 -a 2 ) ; 

 and when these are equal we have 



a 2 = 6 2 (l + sin a) 



where the positive sign is evidently inadmissible, since oar condition 

 above is a less than b. 



Hence from b equal through b equal a to b equal / = , the 



v 1 — sin* 



greatest stress occurs at the extremity of the y axis. After that, the 

 point of greatest stress is nearer to a, but still on the circumference of 

 the ellipse. 



Now if b is very great compared with a, then we have as a 

 limit, 



?/=a,tana, 



T 2 



