﻿388 Mr. L. R. Wilberforce on the Vibrations 



equation Cj- )(P = that we can neglect , ° , and then the 

 above equations reduce to ^ ' 



-iw*i f -a.p^*-^*| + i f » ! 



which may be written 



d 2 

 d 2 



.! 



where a and c are necessarily positive, and b is small. 

 The most general form of solution is 



8x= A x sin^£ + A 2 cos pt + B x sin <?£ + B 2 cos qt, 



8k<j> =■ *-r — ( Ai sin pt + A 2 cos pt) + *-r — (Bi sin qt + B 2 cos qt) , 



where p 2 and q 2 are the roots of the quadratic (a— a) (a—c) — b 2 . 

 We will suppose p 2 to be the greater. The solution may be 

 put into the form 



^~r — 8a — 8kcj> = hi sin {qt + €i) , 



^-r — 8a — 8kcf> = L 2 sin (pi + e 2 ) . 



Thus the motion consists of two normal harmonic vibrations 



of periods — and — ; in the former *-—; — 8a = 8k(f> through- 



or ~~ ci 

 out the motion, and in the latter *-7 — 8a = 8k<f>. 



It is easily seen that p 2 — a is positive and q 2 — a negative ; 

 therefore we conclude that, if b is a positive quantity, the 

 shorter period of vibration corresponds to a screwing motion 

 similar to the screw of the spring, and the longer to a screw- 

 ing motion opposite to the screw of the spring, while if b is 

 negative the reverse is the case. 



It is also clear that if (c—a) is large compared with b and 



«« 53k}. ^ i8 S}. "* ^ is late} « «- 

 in the latter r } CaSe the vibrations of longed} P eriod cor * 



