190 Prof. Challis on the Hydro dynamical Theory of 



however, be said that if the reasoning be such as cannot be 

 called in question, it may possibly suffice to determine, by the 

 consequences to which it leads, whether or not the hypotheses 

 are true. 



1 . As it is proposed to account for the physical forces and the 

 laws of their operation by hydrodynamical pressure, it will first 

 of all be necessary to discuss the principles and rules of the ap- 

 plication of mathematics in hydrodynamics. A perfect fluid 

 (and such, by hypothesis, the ?ether is) may be defined to be one 

 the elementary parts of which possess the properties (1) of being 

 susceptible of movements which continually change their relative 

 positions, (2) of being separable one from another without as- 

 signable force by the insertion of an indefinitely thin solid parti- 

 tion, (3) of pressing against each other and against any solid 

 substance with which they are in contact. In order that the mo- 

 tions of the fluid and its pressures may be capable of mathema- 

 tical treatment, the following axioms must be conceded: — first, 

 that the directions of the motion in any elementary portion of 

 the fluid are always and everywhere normals to a surface which 

 is geometrically continuous through either a finite or an infi- 

 nitely small extent ; secondly, that the motions are consistent 

 with the principle of constancy of mass ; thirdly, that the pres- 

 sures of the fluid, together with the action of extraneous forces, 

 are governed by D^Alembert's Principle. These three axioms 

 being granted, mathematical reasoning founded upon them leads 

 to general differential equations, from the integrals of which 

 may be determined by appropriate treatment the motion and 

 pressure of the fluid under given circumstances. 



2. The first axiom has reference to the principle of geome- 

 trical continuity, to which, it is clear, the motions of the fluid 

 must be subject. Calling, for the sake of shortness, the surface 

 to which the directions of the motion are normal '^ a surface of 

 displacement,^^ it is regarded as an axiom that for each element 

 there is in successive instants such a surface. It is, however, to 

 be consi-dered that a surface of displacement of finite extent may 

 consist of an unlimited number of parts the equations of which 

 are expressed by diff'erent functions, but that neither the tan- 

 gent-planes of two contiguous parts at a given instant, nor the 

 tangent-planes of the surfaces of displacement of a given particle 

 in successive instants, can make a finite angle with each other. 

 These conditions of continuity, which are dynamical rather than 

 geometrical, exclude changes per saltumoi the direction's of mo- 

 tion with respect both to space and time, forasmuch as such 

 changes could only be efi'ected hj infinite forces. 



3. Let, therefore, u, v, iv be the velocities, resolved in the di- 

 rections of the axes of coordinates, at the point of a surface of 



