228 Proceedings of Philosophical Societies. [Sept. 



quantity under the sign is the produce of two factors ; one of 

 "which is an arbitrary function given by the initial state ; the 

 second is a trigonometrical function which has nothing arbitrary 

 in it. This composition of the integral is very worthy of notice, 

 because a great number of physical questions lead to expressions 

 of the same form. 



Analysis separates two parts of the phenomenon, one of which 

 is accidental, and the other constant. The first must be regarded 

 as arbitrary and fortuitous ; it varies in different cases, and 

 the necessary effect of time is to diminish or destroy it; 

 but the second arises only from the single principle of elas- 

 ticity, which is preserved during the whole time of the motion, 

 and is no ways dependent upon the initial figure. 



The final state at which the system necessarily arrives is very 

 simple : it is represented by the trigonometrical function above- 

 mentioned. This consequence does not only agree with the 

 present question ; it is applicable to very different phenomena, 

 the conditions of which are expressed by integrals of the same 

 form. 



The author then goes on to the laws of the motion of the 

 elastic surface, as they are deduced from its integral. A certain 

 part of the surface being at first forced by an external obstacle 

 to depart from its equilibrium, the motion commences as soon as 

 the obstacle is removed. The parts which have not beea 

 removed from the plane of the equilibrium speedily participate 

 in the oscillatory motion, which is immediately propagated far 

 beyond the limits of the original displacement. Three different 

 parts may then be distinguished in the elastic place : one, very 

 iiear the origin, has already ceased to oscillate ; another, which 

 is very remote, has not yet received any sensible agitation ; the 

 second, which is intermediate, is subjected to a motion, which is 

 become regular, and independent of the initial state. The con- 

 centric rings which are formed pass alternately above and below 

 the plane of equilibrium, and at the same time they recede from 

 one another, enlarge, and become lower. The velocity of the 

 summit of each ring is in the inverse ratio of the square of the 

 time elapsed since the origin of the motion ; the distance from 

 one summit to the next is proportional to this square root, the 

 depth of each groove, whether positive or negative, decreases in. 

 an inverse ratio of the time elapsed. 



The author then indicates other motions, an exact idea of 

 which cannot be given without the use of analytical formulae. la 

 the question just mentioned, none of the causes which influence ' 

 the motion has been neglected. Analysis represents, at the 

 «ame time, the forces which determine the first agitations, and 

 those which diminish by degrees the intensity until they render 

 the motion perfectly insensible. It shows how the initial motion 

 m propagating itself into the most remote parts is dissipated, 



