Mathematical Theory of the Vibrations of an Elastic Fluid, 1 35 



as a base for two portions of film at a time. These are the 

 systems of the second class, and are imperfect, because the two 

 portions of film just mentioned are rendered independent of each 

 other by the wire which separates them. Such, for example, are 

 the systems formed by prisms the number of whose lateral faces 

 exceeds six. 



Lastly, with other skeletons we never obtain anything but 

 plane films occupying all the faces of the polyhedron except one. 

 These systems belong to the third class. I call them abortive, 

 because all the films of which they are made up are rendered 

 independent of each other, throughout the whole extent of their 

 perimeters, by the solid wires of the skeleton. I have said that 

 one of the faces remains always unoccupied by a film ; this is 

 because an opening is necessarily required to allow free entrance 

 to the air. Such abortive systems are formed in the skeletons 

 of all the polyhedra, all of whose dihedral angles are consider- 

 ably greater than 120°; I may give as an example the regular 

 icosahedron. 



I conclude the original memoir with the announcement of a 

 further series of researches, in which I shall investigate the 

 systems of films formed by wire skeletons from a different point 

 of view; I shall then show that each of these systems of films 

 takes such a form that the sum of the superficial areas of the 

 films is a minimum. 



XX. The Mathematical Theory of the Vibrations of an Elastic 

 Fluid. By Professor Ciiallis, M.A., F.R.S., F.R.A.S* 



14. HHHE principal object of my last communication was to 

 -i- show that in hydrodynamics three fundamental equa- 

 tions are necessary and sufficient, and to indicate the principles 

 on which the third rests, and the process of investigating it. 

 These three equations may be reduced by elimination to a single 

 differential equation in which ty is the principal variable, and 

 the other variables are x, y, z, and t. The general integral of 

 this equation, if it could be obtained, would not only be appli- 

 cable to particular cases of disturbance of the fluid, but might 

 also give the means of deducing the laws and circumstances of 

 the motion that are independent of arbitrary conditions. But the 

 equation is too complicated to admit of this course being taken ; 

 and happily, so far as regards laws and circumstances of the 

 motion that are not arbitrary, other means are available. Before 

 proceeding to point these out, the following remark, by way of 

 illustration, may be found useful. There is, as is well known, 



* Communicated by the Author. 



