and on the Theory of Tartini's Beats. 23 



continuity of the motion. By using, in accordance with the 

 second of the above principles, this general equation with the . 

 two others that have been long known, I have obtained, prior to 

 any supposed case of disturbance, the following general results: — 

 (1) The small vibrations of an elastic medium, of which the 

 pressure [p) and density (p) are related by the equation ^ = «% 

 in their simplest form are parallel and perpendicular to a recti- 

 linear axis of propagation. (2) Any number of such small 

 vibrations may coexist. (3) The form of the function which 

 expresses the velocity (w) and condensation {s) along the axis of 

 a simple vibration is that of the sine of a circular arCj and the 

 values of v and s are given by the equations 



. Stt , . , \ 



A, 

 (4) To satisfy the conditions of any arbitrary small disturbance, 

 the motion must be supposed to be composed of simple vibra- 

 tions, their number, the directions of their axes, and the values 

 of m, X, and c being arbitrary. 



These results may be at once applied in the undulatory 

 theory of light to account for the kind of vibration which is 

 alone found to accord with phsenomena, for the composite cha- 

 racter of light, and for polarization. Unless these prominent 

 and general facts receive an a priori explanation, we can hardly 

 be said to possess a theory of light. The success with which 

 such explanation is given by the undulatory theory of light 

 based on hydrodynamical principles, may be considered as some 

 evidence that the investigation has taken a right course, and 

 affords a presumption that the same kind of treatment may be 

 applied to the other physical forces. 



There is, however, a case of the motion of an elastic fluid the 

 mathematics of which must be settled before any such applica- 

 tion is possible, viz. that in which the condensation is a function 

 of the distance from a centre, and the motion is in straight lines 

 passing through the centre. This may be called central motion, 

 as forces tending to or from a centre are called central forces. I 

 long since pointed out a difficulty which presents itself in the 

 usual treatment of this case of motion. The known partial dif- 

 ferential equation of the second order between the condensation 

 (s), the distance (r) from the centre, and the time {t),\& satisfied 

 by the solution, 



r 

 from which it necessarily follows, on the principle of the discon- 

 tinuity of the function /, that a single wave of condensation of 

 given breadth may be propagated uniformly from the ccjitrc, 



