1884.] New Spring for Electric and other Instruments. 299 



as the angular twist per unit of length about the axis PTJ, and 



Fr sin x 

 B 



as the angular bending per unit of length abont the axis PS. 



Resolving both of these horizontally and vertically, we have for the 

 total angular motion in a vertical plane, that is, about the axis PM — 



-P, / cos 2 x, . sin 2 a\ 



H~-a- + -b-)' 



and for the angular motion in a horizontal plane — 



Fr sin <x cos a (Jl— JL\ 

 \ A B/ 



If the angular motion, in a horizontal plane, of the free end of the 

 spring relatively to the fixed end be called 0, and if the axial 

 increase of length be called d, and the whole length of the spring- 

 along the spiral Z, then — 



and 



<p=l¥r sin a cos x(——^\ (1). 



V A B/ 



d=lFr*(^+ S ^*) (2). 



\ A B / 



The theory of the strength and stiffness of the ordinary cylindric 

 spiral spring for small angles was given, we believe for the first time, 

 by Professor James Thomson, in the " Cambridge and Dublin Math. 

 Jonrn.," for 1848, and in "Thomson and Tait's Natural Philosophy," 



