1884.] New Spring for Electric and other Instruments. 309 



Under these circumstances, 2a being the major diameter and 2b the 

 minor diameter of the ellipse, 



0= ^^/J__JA ..... (13 ), 



f the maximum stress at a section 



= ^(l + sin a ) (14), 



and d= ZjV;/cos3« 4sin8« \ (15) 



Springs with a 'Rectangular Section. — For practical purposes it is 

 obviously more convenient to use in the construction of our springs 

 thin strips of a rectangular section rather than of an elliptic section. 

 We have, therefore, now to consider how equations 13, 14, and 15 

 will be modified if our strip has a rectangular section, the longer side 

 of the section being 2a and the shorter side 2b. In this case 



and since, as was discovered by M. Cauchy in 1829, the torsion 

 rigidity of a rectangle bears to the torsional rigidity of an inscribed 

 ellipse the proportion of their moments of inertia about a line drawn 

 through the common centre perpendicular to their plane, in the case 

 when one principal dimension is several times the other, it follows 

 that 



a tst 



a " + h " ^ 

 4 



or A=^N—-. 



a? + W 



Hence IFr sin » cos « /_3_ W_3\ 



j ZFr 2 / 3 a 2 + & 2 3 . \ 



a— j COS^ ac + — sir oe. ) ; 



4a6 3 \4N a 2 E / 



or, as b 2 is insignificant in comparison with a 2 , 

 3ZFr sin a cos « 







4a&* 



(i-i) «■* 



3ZFr 2 / cos 2 g sin 2 <* \ ^ 



\ 4N E / K J ° 



d 4 



