Prof. Challis on the Principles of Hydrodynamics. 439 



of motion. The following principle will be found to be of assist- 

 ance in this inquiry : — The general hydrodynamical equations 

 being assumed to be exact and sufficient, any analytical circum- 

 stances which admit of interpretation with respect to the motion 

 prior to the consideration of arbitrary cases of disturbance, have 

 reference to the law of action of the parts of the fluid on each 

 other. 



Proposition X. It is required to determine the law of the 

 mutual action of the parts of a compressible fluid, the pressure 

 of which varies in the same proportion as the density. 



(1.) The following equation was obtained in the proof of Pro- 

 position VI. (Phil. Mag. for January 1851, p. 33), viz. 



X, [dyp') = udx + vdy + wdz. 



Now by an abstract theorem of analysis, the right-hand side of 

 this equality is integrable if X be a function of -v/r, or more gene- 

 rally, a function of <f and t. The same quantity is integrable 

 in an unlimited number of ways by particular values of u, v, and 

 w, depending on particular arbitrary disturbances. But the 

 supposition that X is a function of i|r and Hs of a general natui-e, 

 and may be made prior to the consideration of any case of 

 motion. Hence, according to the principle above enunciated, 

 if this supposition conducts to a result compatible with fluid 

 motion, that result is indicative of the mode of action of the parts 

 of the fluid on each other. But by Proposition VII. it was 

 shown, that if X be a function of i/r and t, the motion is recti- 

 linear. Consequently, if the mode of action of the parts of the 

 fluid on each other be such as to satisfy the condition of making 

 vdx + vdy + wdz integrable, the motion is rectilinear. 



At this stage of the reasoning it will be necessary to refer to 

 the results which were obtained in the January Number (1851), 

 by a consideration of rectilinear motion perpendicular to a fixed 

 plane, and rectilinear motion tending to or from a fixed centre. 

 (Examples I. and II. p. 34-37.) In each of these cases of motion 

 absurd results were arrived at by reasoning strictly in accordance 

 with the received principles of hydrodynamics. As those prin- 

 ciples arc not untrue, it hence follows that they ai-e insufficient 

 for the solution of hydrodynamical problems. Also, as the con- 

 tradictory results were deduced from true principles, it is certain 

 that the reasoning involved some false step, which it is essential 

 to discover. Where the error was committed will appear in the 

 course of the following investigation. 



If the motion be in directions perpendicular to a fixed plane, 

 and be a function of the distance from the plane, it will be recti- 

 linear motion, and will satisfy the condition of making udx + vdy 

 -\-wdz integrable. May we, therefore, sup])Osc that the parts 



