174 Prof. Challis on the Hydrodynamical Theory 



more accurately and completely in pages 201-228 of i The 

 Principles of Mathematics and Physics,' I have obtained par- 

 ticular solutions of the general hydrodynamical equations ap- 

 plicable to the motion of a fluid for which p = d 2 p, without 

 having made previously any supposition as to the circum- 

 stances under which the fluid was put in motion. These re- 

 sults, as depending on no special arbitrary conditions, but only 

 on the general analytical hypothesis that udx + vdy + ivdz is 

 an exact differential, are considered to give the laws of spon- 

 taneous mutual action between the parts of the fluid. The 

 motion thence resulting is characterized by being symmetrical 

 with respect to an axis, and by consisting wholly of harmonic 

 vibrations partly parallel and partly transverse to the axis. 



6. It is not necessary to introduce here the details of the 

 abstract reasoning whereby this particular motion was reached ; 

 but it is important to remark that as the solution is unique 

 and definite it cannot without error be left out of consideration 

 in applications of general solutions of the hydrodynamical 

 equations, and that it must be taken into account before pro- 

 ceeding to apply such solutions to cases of motion taking 

 place under given arbitrary circumstances. This being pre- 

 mised, I shall now adduce the investigation of that particular 

 motion in sufficient detail for the present purpose ; and in 

 order to give the means of acquiring fuller information, re- 

 ferences will be made to the pages of the above mentioned 

 work, or to Numbers of the Philosophical Magazine, in which 

 the several steps of the reasoning are completety discussed. 



7. A differential equation, which I call the first general 

 equation of hydrodynamics, being formed on the principle 

 that the directions of motion in each given element are normal 

 to surfaces of displacement in successive instants, it may be 

 shown that if udx + vdy + wdz be an exact differential (dyjr), 

 the motion is spontaneously rectilinear where yfr has a maxi- 

 mum or minimum value (Props. VII. and XI. in pages 186 

 and 201 ; the reasoning is given better in arts. 32-35 of the 

 communication in the Philosophical Magazine for September 

 1872). Accordingly, assuming that there would be rectilinear 

 motion along an axis, it may be supposed, for the rest of the 

 motion, that (d ■^) — {d.f <£), and, the ordinates z being 

 reckoned along that axis, that / is a function of x and y only, 

 and <j> a function of z and t only. These assumptions are 

 justified by actually finding definite expressions for the 

 functions <f> and / by purely analytical reasoning. It is found, 

 in fact, by such process (pages 201-205 and 210) that, to the 

 first approximation, 



