1884.] New Spring for Electric and other Instruments. 307 



fifths of Young's modulus, the ratio given by M. St. Yenant, this value 

 of tan a. becomes equal to V /, 625, or 



*=38° 19', 



makes — a maximum. 

 d 



We have already seen, from (3), that to merely make a maximum 

 for a given axial force, a. ought to be 45°, and I and r as great as 

 possible. 



We are therefore led from three considerations — 



1. That an axial force shall produce a maximum relative rotation ; 



2. That the rotation shall be great without producing permanent 

 set in the material ; 



3. That the rotation shall be great in comparison with the elonga- 

 tion of the spring ; 



to make the angle of the spiral about 45°. 



Further, to produce the first two of these conditions, it will be 

 observed that the length of the wire forming the spring ought to be 

 great, while the third condition is independent of the length. 



Next, with regard to the proper radius r to give to the coils of the 



spring. increases with r, - is independent of r, and - varies in- 



/ * 

 versely as r. Hence, as the first and third conditions are antago- 

 nistic one to the other, the value of r must be chosen to suii the 

 conditions of the instrument in which the spring is to be used ; that 

 is, we must consider in any special case whether the possibility of 

 permanent set or a large axial motion is more to be avoided in the 

 particular instrument in question. 



Let us now consider how 0, -, and - depend on a and b. Refer- 



/ d 



ring to equation (3), if b is not greater than a, it is clear that the 

 smaller b is the larger will be. Next putting N equal to f E, an 

 approximation sufficiently accurate for the substances likely tc be 



employed, we see from equation (10) thaty depends on 



~w( 1-5 ~l§) 



and as in this first case we are limited to values of b between and 



a 



Sin ex 



or between and about 1*84 a if we put a equal to 45°, 



a value not far from that which we have already determined to be the 

 best, then it is obvious that the smaller b is the larger will be 



