Modes of a Vibrating Si/stem. 435 



longs to the latter class. It is only by the introduction of limi- 

 tations, such as attributing to various parts absolute rigidity, 

 that the position of the whole is reduced to dependence on a 

 finite number of coordinates. But even after every permissible 

 simplification has been made, there remains a large and import- 

 ant class of systems whose configuration cannot be defined with- 

 out the specification of an infinite number of coordinates, or, 

 which comes to the same, of a function of one or more indepen- 

 dent variables. Under this head must be included all strings, 

 bars, membranes, plates, &c. which are treated as capable of 

 continuous deformation. 



To each fundamental mode corresponds what may be called a 

 normal function. To determine these in the case of any parti- 

 cular system is a problem which may tax, and will usually over- 

 tax, all the power of analytical expression which the mathemati- 

 cian possesses ; but whether expressed in terms of simple func- 

 tions or not, the normal functions must be thoroughly discussed, 

 not to say tabulated, before the solution of the problem can be 

 considered complete. 



The normal functions appear analytically as the solution of a 

 differential equation containing a constant at this stage unde- 

 termined ; and the first step is the formation of this charac- 

 teristic equation. The usual method proceeds by the considera- 

 tion of the forces actually acting on an element in virtue of its 

 connexion with the rest of the system. For example, the ele- 

 ment of a flexible string is acted on by the tensions at its two 

 extremities ; and the equation of motion expresses the fact that 

 the actual acceleration of the element is proportional to the re- 

 sultant of the tensions when resolved in the transverse direction. 

 The characteristic equation is obtained on the introduction of the 

 assumption that the whole motion is simple harmonic. 



The second method, which was (I believe) first employed by 

 Green, depends on the use of what we now call the potential 

 energy ; and my present object is to point out its advantages. 

 For this purpose I will take, as neither too easy nor too difficult, 

 the problem of the transverse vibrations of a thin uniform rod 

 whose natural condition is straight. 



The potential energy V is for each element of length propor- 

 tional to the square of the curvature ; and thus, if y denote the 

 transverse displacement of the element whose distance from one 

 end of the rod is x, we have 



Y<&, ...... 



-»«J©) 



where the integration must extend over the length of the rod, 

 from to /. 



2 G2 



