System about a Configuration of Equilibrium. 451 



(b) The Fourier series expressive of each coordinate con- 

 tains cosines only, without sines, of the multiples of nt. 

 Thus the whole system comes to rest at the same moment of 

 time, e.g. t = 0, and then retraces its course. 



(c) The coefficient of cos rnt in the series for any co- 

 ordinate is of the rth order (at least) in the amplitude (H 2 ) 

 of the principal term. For example, the series of the third 

 approximation, in which higher powers of H^ than H^ are 

 neglected, stop at cos 3nt. 



(d) There are as many types of solution as degrees of 

 freedom ; but, it need hardly be said, the various solutions 

 are not superposable. 



One important reservation (it was added) has yet to be 

 made. It has been assumed that all the factors, such as 

 (c 2 — 4tt 2 a 2 )*, are finite, that is, that no coincidence occurs 

 between an harmonic of the actual frequency and the natural 

 frequency of some other mode of infinitesimal vibration. 

 Otherwise, some of the coefficients, originally assumed to be 

 subordinate, become infinite, and the approximation breaks 

 down. 



I have lately had occasion to consider more closely what 

 happens in these exceptional cases ; and I propose to take as 

 an example a system with two degrees of freedom, so con- 

 stituted that the frequencies of infinitesimal vibration are 

 exactly as 2 : 1. In the absence of dissipative and of im- 

 pressed forces, everything may be expressed by means of the 

 functions T and V, representing the kinetic and potential 

 energies. In the case of infinitely small motion in the 

 neighbourhood of the configuration of equilibrium, T and V 

 reduce themselves to quadratic functions of the velocities 

 and displacements with constant coefficients, and by a suit- 

 able choice of coordinates the terms involving jyroducts of the 

 several coordinates maybe made to disappear. Even though 

 we intend to include terms of higher order, we may still 

 avail ourselves of this simplification, choosing as coordinates 

 those which have the property of reducing the terms of the 

 second order to sums of squares. We will further suppose 

 that T is completely expressed as a sum of squares of the 

 velocities with constant coefficients, a case which will include 

 the vibrations of a particle moving in two dimensions about 

 a place of equilibrium. We may then write 



T=ia 1 ^ 1 2 +ia 2 ^ 2 2 , (1) 



V = ie 1 <£ 1 2 + i^ + V 3 + ...., . . (2) 



* See below. 



2H2 



