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Profs. W. E. Ayrton and J. Perry. [Feb. 14, 



The twisting torque to which the spring is subjected is Fr cos a, 

 and the bending torque to Fr sin a. But the twist must be multi- 

 plied by sin a, and the bend by cos a when we project these motions 

 on a horizontal plane. So far then as the total rotation in a horizontal 

 plane of the free end of the spring relatively to the fixed end is con- 

 cerned, it may be regarded as being produced by equal twisting and 

 bending torques, each of them equal to Fr sin x cos a. ; and the total 

 rotation of the free end of the spring relatively to the fixed end, which 

 is the special feature of the springs we are considering, is proportional 

 to the difference between the two angular rotations produced in the 

 wire by these equal bending and twisting torques. The twist alone 

 would cause an increase in the number of coils, that is, a rotation in the 

 direction of coiling which is what we call positive, while the bending, 

 or rather the unbending, alone would cause a negative rotation, or 

 one tending to uncoil the spring. When both occur together in the 

 actual spiral spring subjected to an axial force the total rotation is 

 positive or negative, according as the angular twist or the angular 

 bend is the greater. Hence the flexural and torsional rigidities of the 

 wire alone determine whether the rotation is positive or negative. 



It is well known, for example, that when a wire of circular section 

 is subjected to equal twisting and bending torques the twist is greater 

 than the bending for almost all substances, that is, substances in 

 which the ratio of the modulus of rigidity to Young's modulus is 



