Vibrations of Heavy Bodies. 101 



In the next place, regarding friction as the sole retarding 

 power which is known to act simply in proportion to the time, 

 and without any reference to velocity. 



It is obvious that while this is supposed to be extremely 

 small in comparison with the force of gravity, resolved into 

 the direction of motion at the commencement of the descent, 

 and all increase of weight in the oscillating body arising from 

 centrifugal force, is disregarded, as being insensible ; that the 

 retardation of velocity in isochronous vibrations must be equal. 



If this general deduction, however, admits of doubt, it may 

 be demonstrated in the following manner : 



-? 



The velocity at nr (Fig. 2d,) will be "'~^; consequently 



the time of passing through 2i will be — ' ^ ' ^ . Let the 



uniformly retarding power of the friction, as compared with the 

 constant force of gravity be g, then will the fluxion of tlie re- 

 tardation be -v '^ LJ^the fluent of which is 

 Go — x^i 



H . ^ 2 . g y. Cir. Arc to radius anity and cos. — 



a 



When .r = a 



■=■ 3. . ^2 . g X quadrantial Arc to ra'dius unity, which is 

 a constant quantity. 



Since, then, the velocities are uniformly diminished, so will 

 be the arc of ascent due to such velocities, from what has been 

 already shewn : assuming therefore, as before, a, to be the in- 

 cipient semi-arc of free vibration, and b equal to the final semi- 

 arc, the time of passing from one to the other to be unity, x an 

 elapsed portion of that time, and y the corresponding arc of 

 semivibration with m a modulus, 



X — — my the fluent x ——my -f o, when a;=:0 c=wia 

 The whole fluent, therefore, x=ma — my, when x=:lw=i, 



consequently 1 = ma — mb, or m =^ — whence 



a — b 



H 2 



