160 OF DEFLECTIVE FORCES. 



combined: then since vzz—ff^ty and s —fvdit, we have 



v:=. — / d^=: — cos f + c, and «= — sin ^ + cf == a 



^ m m m 



sm f-f sm ^, and c=0, — =: a + 

 rti 



l,a= 1: (— -l) = 



m .„ 1 1 - „ 



or II 71-=. — , 7, as beiore. 



Scholium 4. If the oscillating body be initially in 

 any otlier condition, its subsequent motion may be deter- 

 mined, by considering it as performing a secondary vibra- 

 tion with respect to a point vibrating in the manner here 

 supposed, which will consequently represent its mean 

 place ; but if there be no resistance, the body will have 

 no tendency to assume the form of a regular simple vibra- 

 tion, rather than any other. Supposing, for example, that 

 the point had been initially at rest in the middle vertical line, 

 when the centre of suspension passed that line ; it will then 

 agree in situation with the point representing its mean 

 place, but not in velocity ; and it will return to its mean 

 place after every interval equal to a complete single spon- 

 taneous vibration of the true pendulum ; and when this 

 coincidence happens in the middle vertical line as at first, 

 the whole cycle of motions will begin again, after a period 

 depending on the comparative lengths of the supposed 

 pendulums : and at some intermediate time the coincidence 

 will in most cases occur near the extremity of the vibra- 

 tion representing the mean place, and the excursion will 

 be much greater than that of this vibration, while at ano- 

 ther part of the cycle it may be almost obliterated. Such 

 a succession of cycles may be often observed in the actual 

 vibrations of elastic bodies of irregular forms, the excur- 

 sions being alternately greater and smaller without any 

 interference of external causes. 



