144 THIRD REPORT — 183.'i. 



circles round any disturbance made at the surface of water. 

 No theory of waves which does not embrace these can be con- 

 sidered complete. In the essay of M. Cauchy, which obtained 

 the prize, and is printed in the M^moires des Savans*, the 

 theory of only the first kind of waves is given. This essay, 

 however, claims to be more complete than the first part of 

 M. Poisson's memoir, because it leaves the function relative to 

 the initial form of the fluid surface entirely arbitrary, and conse- 

 quently allows of applying the analysis to any form of the body 

 immersed to produce the initial disturbance. M. Poisson re- 

 stricts his reasoning to a body, of the form of an elliptic para- 

 boloid, immersed a little in the fluid, with its vertex downward 

 and axis vertical ; and as this form may have a contact of the 

 second order, with any continuous surface, the reasoning may 

 be legitimately extended to any bodies of a continuous form, 

 but not to such as have summits or edges, like the cone, cy- 

 linder and prism. This restriction having been objected to as a 

 defect in the theoryf , M. Poisson answers J that his analysis 

 is not at fault, but that one of the differential equations of the 

 problem, which expresses the condition that the same particles 

 of water remain at the surface during the whole time of motion, 

 very much restricts the form which the immersed body may be 

 supposed to have. When the initial motion is produced by the 

 immersion of a body whose surface presents summits or edges, 

 it is not possible, he thinks, to represent the velocities of the 

 fluid particles by analytical formulae, especially at the first in- 

 stants of the agitation, when the motion must be very complicated, 

 and the same points will not remain constantly at the surface. 



With the exception of the particular we have been mention- 

 ing, the two essays do not present mathematical processes es- 

 sentially different in principle. Attached to that of M. Cauchy, 

 which was published subsequently to M. Poisson's memoir, are 

 Valuable and copious additions, serving to clear up several 

 points of analysis that occur in the course of the work, and re- 

 ferring chiefly to integration by series and definite integrals, 

 and to the treatment of arbitrary functions. Among these is a 

 lengthened discussion of the theory of the waves uniformly 

 propagated, the existence of which, as indicated by the analysis, 

 had escaped the notice of both mathematicians in their first re- 

 searches. In this discussion the velocities of propagation are 

 determined of the two foremost wajVes produced by the immer- 



* vol. iii. 



t Bulletin de la Societe Philomatique, Septembre 1818, p. 129. 

 J "Note sur le Probleme des Ondes," torn. viii. of Memoires de I'Academie 

 des Sciences, p. 571. 



