158 Mr. G. G. Stokes on the alleged necessity for 



pei'iiiission, I now enter, on the following plan : to notice only 

 one important point at a tintie ; and if a question should arise 

 which Professor Challis and I both regard as of vital importance, 

 but on which we cannot agree, to leave off the discussion. 



The form which Professor Challis has given to his last com- 

 munication, by commencing at the beginning of the subject, 

 dispenses with the necessity of referring to previous publications 

 or controversies, and I shall therefore proceed at once to the 

 discussion. 



The first five propositions in Professor Challis's paper contain 

 only what is generally admitted by mathematicians, and which 

 I admit among the number. On this part of the paper I will 

 only make three remarks. First, that although the equality of 

 pressure in all directions follows in a similar manner fi'om the 

 fundamental principle assumed in the ordinary theories of hy- 

 di'odynamics and hydi-ostatics, I do not conceive that the funda- 

 mental principle is equally accurate, experimentally, in the two 

 cases; secondly, that I consider it clearer and more coi-rect to 

 regard the pressure as connected with the forces, &c. by three 

 equations (partial differential equations of course), than by the 

 single equation numbei'ed (1.) in Professor Challis's paper; 

 thirdly, that I conceive there is a slight, but not unimportant, 

 omission in Professor Challis's demonstration of the ordinary 

 equation of continuity. I make these remarks merely to guard 

 my statement in assenting to all that is contained in the first 

 five propositions, and not with a view of exciting controversy, and 

 therefore I do not support my opinions by any arguments. 



The point which I select for discussion in the present com- 

 munication is the second axiom : " let it be granted that the di- 

 rections of motion in each element of the fluid mass may at all 

 times be cut at right angles by a continuous surface." After 

 enunciathig this axiom, Professor Challis remarks, '^by satis- 

 fying this geometrical condition, the motion of a continuous mass 

 is distinguished from that of a collection of indefinitely small 

 discrete atoms." This assertion, which is altogether unsup- 

 ported by any argument, I contend to be erroneous. I shall 

 bring forward two arguments in support of my opinion ; not 

 because I conceive that either would not be abundantly sufficient 

 by itself, but because, while the first is rather the simpler, the 

 second will give rise to a remark of some importance. 



First argument. Let us consider the motion for which 



M= — (oy, v=-wx, w=c, .... {a.) 



where w and c are constant. If the lines of motion admit of 

 being cut at right angles by a system of sui'faces, the differential 

 equation to the system must be udx + vdy + wdz — 0, and udx + &c. 



