14 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 



initial velocities by differentiating with respect to t, and replacing the arbitrary functions which 

 represent the initial velocities by those which represent the initial displacements. The same 

 result constantly presents itself in investigations of this nature : on considering its physical 

 interpretation it will be found to be of extreme generality. 



Let any material system whatsoever, in which the forces acting depend only on the posi- 

 tions of the particles, be slightly disturbed from a position of equilibrium, and then left to 

 itself. In order to represent the most general initial disturbance, we must suppose small 

 initial displacements and small initial velocities, the most general possible consistent with the 

 connexion of the parts of the system, communicated to it. By the principle of the superposi- 

 tion of small motions, the subsequent disturbance will be compounded of the disturbance due 

 to the initial velocities and that due to the initial displacements. It is immaterial for the truth 

 of this statement whether the equilibrium be stable or unstable ; only, in the latter case, it is 

 to be observed that the time t which has elapsed since the disturbance must be sufficiently small 

 to allow of our neglecting the square of the disturbance which exists at the end of that time. 

 Still, as regards the purely mathematical question, for any previously assigned interval t, how- 

 ever great, it will be possible to find initial displacements and velocities so small that the 

 disturbance at the end of the time t shall be as small as we please ; and in this sense the prin- 

 ciple of superposition, and the results which flow from it, will be equally true whether the 

 equilibrium be stable or unstable. 



Suppose now that no initial displacements were communicated to the system we are con- 

 sidering, but only initial velocities, and that the disturbance has been going on during the time 

 t. Let/(£) be the type of the disturbance at the end of the time t, where f(t) may represent 

 indifferently a displacement or a velocity, linear or angular, or in fact any quantity whereby 

 the disturbance may be defined. In the case of a rigid body, or a finite number of rigid 

 bodies, there will be a finite number of functions f(t) by which the motion of the system will 

 be defined : in the cases of a flexible string, a fluid, an elastic solid, &c, there will be an infinite 

 number of such functions, or, in other words, the motion will have to be defined by functions 

 which involve one or more independent variables besides the time. Let « be in a similar 

 manner the type of the initial velocities, and let t be an increment of t, which in the end will 

 be supposed to vanish. The disturbance at the end of the time t + t will be represented by 

 fit + t) ; but since by hypothesis the forces acting on the system do not depend explicitly on 

 the time, this disturbance is the same as would exist at the end of the time t in consequence of 

 the system of velocities v a communicated to the material system at the commencement of the 

 time — t, the system being at that instant in its position of equilibrium. Suppose then the 

 system of velocities « communicated in this manner, and in addition suppose the system of 

 velocities - v communicated at the time 0. On account of the smallness of the motion, the 

 disturbance produced by the system of velocities v will be expressed by linear functions of 

 these velocities; and consequently, if/(£) represent the disturbance due to the system of velo- 

 cities v , - f(t) will represent the disturbance due to the system - v . Hence the disturbance 

 at the end of the time t will be represented by f[t + t) -f(t). Now we may evidently 

 regard the state of the material system immediately after the communication of the system 

 of velocities - v as its initial state, and then seek the disturbance which would be produced 

 by the initial disturbance. The velocities v going on during the time t will have produced 



