142 



Major Hersckel. On a Simplified Form of the [Dec. 9, 



horizontal direction, tending to cause it to return to its position of 

 rest, may be expressed by 



P . — sin -7- a / 1— versm 



R » R 



exerted at a distance r from the axis of motion. When — is small, 



R 



this may be represented by the simple function P-^sin#, which varies 



R 



directly as the sine of the angle of detortion, being at zero and 

 180°, and a maximum at 90°. 



This is the ordinary law of torsion of a bifilar balance, in which the 

 suspending lines are regarded as non-resisting. 



Now suppose the weight P held fast at its position of rest while 

 the upper ends of the suspending lines, no longer impotent, but 

 endued with elasticity and a consequent power of resistance to torsion, 

 are turned, severally, about their own individual axes, through an 

 angle rj. The force which will be thereby developed, in what we may 

 now call the wires, will be a true " torsion " force. And, if I am 

 not mistaken, it will vary directly as the angle of torsion, and 

 inversely as the length of the wire, but not as the tension nor as the 

 distance of the wires from each other. 



Let T be what I may call the factor of torsion, for the particular 

 quality of wire in use. By this ± mean the force, measured in grains 

 (provisionally), which, exerted at a distance unity (in inches) from 

 the axis, will balance the tendency of the wire, when twisted through 

 one turn to each unit of length, to untwist. Then, under the actual 

 circumstances, the torsion of the pair will be expressed by 



o 1 T 



2 ^'R ; 



which must be divided by r to denote its power, applied at the ends of 

 the wires, to turn the system. 



Now, let P be released, so that this torsion may act upon it. Let 

 7) — 9-\-<fi ; then will take the place of rj in the last expression, and the 

 forces in opposition will be 



P IT 



— r sin 6, and — . — ■ . ; 



R TTY R 



and 9 will obviously increase and decrease until there is equilibrium. 

 Hence the approximate statical equation of the torsion-balance 



Pr sin0=J^T0, 



I T 



orsm#=A0, if A= — -. — 



77T 3 P 



