26 Prof. Challis on Newton's 



form of the atom, but only as a ground for mathematical rea- 

 soning whereby the truth of the hypothesis may be tested. 



Before proceeding to Part (II.) of the stage of physical 

 inquiry now under consideration, a certain mathematical diffi- 

 culty has to be stated and cleared up. In physical astronomy, 

 as we have seen, Newton overcame the difficulty of discover- 

 ing the calculation proper for determining the motion of a 

 single particle acted upon by given forces. Since, according 

 to the foregoing hypotheses, the pressure of the astherial me- 

 dium performs an essential function, we have now to discover 

 the hydrodynamical principles and calculations applicable to 

 the determination of the motion and pressure of a fluid par- 

 ticle in juxtaposition with other particles, which is a prelimi- 

 nary difficulty of the same kind as that just mentioned. The 

 former process required the formation of a differential equation 

 containing two variables ; this requires differential equations 

 containing at least three variables, and is therefore of greater 

 complexity. The discovery and treatment of two such equa- 

 tions, one expressing the principle of constancy of mass, and the 

 other derived from an application of D'Alembert's Principle, 

 were effected by the researches of Euler, Lagrange, and 

 Poisson. On attempting to advance to Part (II.) by employ- 

 ing these equations, I was stopped by finding that Poisson 

 had deduced from them, in the case of a fluid the pressure of 

 which varies proportionally to its density, an equation from 

 which I could strictly infer that the same fluid particle might 

 be at rest and have a maximum velocity at the same moment 

 of time (see Phil. Mag. for June 1848, p. 496, and my ' Prin- 

 ciples of Pure and Applied Calculation/ p. 195). This was 

 evidently a reductio ad absurdum absolutely demanding a re- 

 consideration of the hydrodynamical principles. As those 

 principles on which the two equations are founded are unques- 

 tionably true, the only logical inference is that they are in- 

 sufficient. It consequently occurred to me that, besides the 

 equation expressing constancy of mass, one was required for 

 expressing continuity of the motion. Accordingly I have pro- 

 posed to found such an equation on the following principles : — 

 (1) that the course of a given particle is so far continuous 

 that the directions of its motion in successive instants cannot 

 make with each other a finite angle ; (2) that the directions 

 of the lines of motion in each elementary particle are at each 

 instant normals to a surface of continuous curvature, which is 

 either of finite or infinitely small extent, the total surface of 

 displacement being conceived to be such that, however it 

 might be composed of discontinuous parts, no two of its con- 

 tiguous elements are inclined to each other by a finite angle. 



