1884.] New Spring for Electric and other Instruments. 303 



between one-third and one-half. Hence we may expect that in a 

 spring made of round wire, and with the spires making an angle of 

 45° with a plane perpendicular to the axis, the total rotation will 

 be positive for an axial force applied so as to lengthen the spring. 

 And experiment shows that this is the case. 



If the wire be flattened and bent so that the flat side of the strip 

 touches the cylinder on which the wire is coiled, as shown in fig. 3, 

 then the arrangement is such that the bending is greater than the 

 twist. Hence an axial force applied so as to lengthen this spring 

 causes a negative rotation, whereas if the strip be coiled as in fig. 4, 

 so that the edge of the strip lies against the cylinder on which it is 

 coiled, an axial force similarly applied will now cause a positive rota- 

 tion. It is almost certain that for any strip of material the positive 

 value of obtained with the latter form of spring is likely to be 

 greater than the corresponding negative value with the former kind, 

 but the difficulty of manufacturing the second form of spring, where 

 h has a very small value, has compelled us to confine our attention to 

 the former type. 



Permanent Set. — Having constructed some very delicate springs of 

 this kind, one of the first difficulties which we met with arose from 

 their liability to acquire a permanent set, so that it has been necessary 

 to determine the dimensions of the spring which will give the largest 

 amount of rotation with the minimum amount of stress in the 

 material. 



The shear stress at any point x, y in an elliptic section if the prism 

 has received a twist t per unit length, is — 



where a and b are the principal semi- diameters of the ellipse, the rxis 

 of y being parallel to the semi-diameter b. 

 Consequently these shear stresses equal — 



and 



where 



2JfNrX 



a 2 + 6 3 



and 



a 2 + 6 2 



respectively ; 



and the total shear stress is equal to 



2Nt 



VOL. xxxvi. 



T 



