OF DEFLECTIVE FORCES. 159 



and when n — 1 is negative, the displacement being in a 

 direction opposite to that of the supposed point of sus- 

 pension. Consequently, when a body is performing oscil- 

 lations by the operation of any force, and is subjected to 

 the action of any other periodical forces, we have only to 

 inquire at what distance a moveable point must be situated 

 before or behind it, in order to represent the actual mag- 

 nitude of the periodical force by the relative situation, 

 according to the law of the primary force concerned, and 

 to find an expression for this distance in terms of the sines 

 of arcs increasing equably, in order to obtain the situation 

 and velocity of the body at any time, provided that we 

 suppose it to have attained a permanent state of vibration. 



Scholium 3. We may easily express this reasoning in 

 a form more strictly algebraical : thus the time, with respect 

 to the forced vibration of the centre of suspension, being 

 called t, the place of the vertical Hne passing through that 

 point will be indicated by sin f, supposing t to begin from 

 the middle of a vibration: now the force acting on the 

 moving body will always be as its distance from this 

 moveable vertical line, considered with relation to the 

 length of the true pendulum m ; that is, it will be expressed 



c sin t 



by/= , the unit of m being the length of the imagi- 

 nary pendulum carrying the point of suspension, since when 

 s-=zO and sin t—l, the force must bezzl orizo-. Now we 

 may satisfy this equation by the particular solution s — sin t 

 =:a sin t, which represents a vibration either correspond- 

 ing in its direction with the former, or in an opposite 

 direction, accordingly as a is positive or negative ; and s, 

 the space actually described, will be the sum or difference 

 of the spaces belonging to the separate vibrations so 



