228 Prof. J. D. Everett on Dynamical 



Lectures ; and in this connexion I have hit upon a curious 

 geometrical and kineinatical theorem which I believe is new. 



In dealing with mutually influencing pendulums, whether 

 suspended side by side or one from the other, I have simplified 

 the usual investigation, by first assuming the motion to take 

 place in a fundamental mode, and afterwards discussing com- 

 binations of such modes. The deduction of the properties of 

 the double pendulum here given is, I believe, the fullest to 

 be found anywhere. 



A " fundamental mode of vibration " may be defined as 

 one in which all the particles have simple harmonic vibrations 

 mill the same period and either identical or opposite phases. 

 The resolvability of the most general small permanent vibra- 

 tions of a system into modes of this simple character is a 

 well-known proposition of abstract dynamics. Anyone who 

 objects to assuming it may regard our plan of operations as 

 a tentative method which is justified when it leads to solutions 

 containing the proper number of arbitrary constants. 



§ 2. Suppose a series of equal particles attached at equi- 

 distant points to a uniform elastic string of negligible mass. 

 For simplicity we shall ignore gravity, and suppose the only 

 forces acting on each particle to be the tensions of the two 

 .portions of string between which it lies. Then the series, 

 when in equilibrium, will lie in a straight line, with uniform 

 tension throughout the whole string. Let F denote this 

 uniform tension, and a the common distance between the 

 particles. Then, in the case of small transverse vibrations in 

 one plane, if y^y^y?, denote the displacements of three con- 

 secutive particles, and the mass of each particle be M, we 

 shall have 



y *~ M a M a > 



or, putting \x for F/Ma, 



h= -^(#2-yi)--K2/2-#3)= -M^a-yi-ys). • (1) 



These formulae are also applicable to longitudinal vibrations, 

 if we make F denote Young's modulus multiplied by the 

 cross-section. 



§ 3. When a simple harmonic undulation is running along 

 the chain of particles, we shall have 



y=A sin 2^-1), (2) 



y denoting the displacement of the particle whose undisturbed 

 abscissa is #, \ the wave-length, t the time, T the period, 

 and A the amplitude. 



