164 Of DEFLECTIVE FORCES. 



Scholium 1. Supposing the relation of the resistance 

 to the velocity to be altered, the relation of the sine AC to 

 the cosine CD must be similarly altered, the force equiva- 

 lent to the resistance varying as the sine, and the extent 

 of the vibrations, and consequently the velocity, a« the co- 

 sine of the displacement BI : but the relation of the sine 

 to the cosine is that of the tangent to the radius : so that 

 the tangent of the displacement will be as the mean resist- 

 ance : and the sine of the displacement, AC, is to the ra- 

 dius BC, as the greatest resistance is to the greatest force 

 which would operate on the pendulous body if it remained 

 at rest at G: the displacement at the extremity of the 

 vibration having the same angular measure, but becoming, 

 with respect to the place of the body, the verse sine only, 

 instead of the sine. 



Scholium 2. It is obvious, from the figures, that the 

 body G will always be behind the place S, which it would 

 have occupied without the resistance, when the vibration is 

 direct, but before it when inverted. 



Scholium 3. When the resistance is very small, a 

 simple pendulum with a similar resistance may be conceived 

 to vibrate nearly in a similar manner: and if we neglect the 

 diminution of the velocity in the consideration of the re- 

 sistance, and call rzzmv=.m cos t, we have v=.—Jfditiz 

 —f{sm t + m cost)dt=:cos t — m sin t, and s^zjvdtzz sin t 

 ^m cos t — a=i -v/ (1 4-^2) sin (f + &) — a, b being the angle 

 of which the tangent is m (216), and azz^/{l +m^) sin hzz 



^(l-^-rriP) ; zifw, consequently s:=z ^J(\-\-w?) sin 



/v/(l-{-mw) 



it + h) — vif which implies a vibration observing the period 



of t, but beginning at ^ point at the distance h further back 



in the circle, so that the time of ascent will be diminished 



and that of djescent increased very nearly in an equal de- 



