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LX. On the general Differential Equations of Hydrodynamics, 

 By Professor C hall is, F.R.S.* 



1. r I^IIE propositions in hydrodynamics, the proofs of which 

 J- I recently expressed the intention of bringing under 

 review, are contained for the most part in communications to 

 the Numbers of the Philosophical Magazine for January 1851, 

 March 1851, December 1852, and February 1853. In the 

 references that will be made to these communications, the 

 meanings of the symbols will be supposed to be known ; and in 

 the present one the same symbols will be used, and in the same 

 significations. The article 4f 0n the Principles of Hydrody- 

 namics^ in the Number for January 1851 contains definitions 

 of two fundamental properties of a perfect fluid, and the proofs 

 of six propositions founded on these properties, and on self- 

 evident principles. The first five of the propositions need not 

 be particularly dwelt upon, as the reasoning by which they are 

 established is not new, and has been generally accepted. Re- 

 specting the fundamental properties, viz. that the parts of a fluid, 

 press mutually and against the surface of a solid, and that, if the 

 fluidity be perfect, the parts are separable by an infinitely thin 

 partition without assignable force, I will only remark that as 

 they are obvious and distinctive, and rest on experimental 

 evidence, they seem to be the most appropriate that can be 

 thought of for the basis of mathematical reasoning applied to 

 fluids. The proofs of Propositions I. and II. based upon them, 

 the one demonstrating the law of pressure in the case of equili- 

 brium, and the other the same law in case of motion, must be 

 considered to be as exact, on the hypothesis of perfect fluidity, as 

 are those proofs of propositions in statics and dynamics which 

 rest on the hypothesis of the perfect rigidity of solids. Also 

 the law r of pressure is as strictly proved for fluid in motion as 

 for fluid at rest. 



2. Proposition VI., which has reference to a new general 

 differential equation, will require more particular consideration, 

 since it cannot be expected that such an equation will be ad- 

 mitted except upon ample evidence of the necessity for it, and 

 of its truth. I propose, therefore, to devote this communication 

 mainly to the discussion of the circumstances which render 

 necessary a third general hydrodynamical equation, and of the 

 process by which it may be investigated. 



3. Ecfore entering upon this inquiry, it will be proper to 

 adduce the two commonly received hydrodynamical equations, 

 and to state briefly the principles on which they rest. The first 

 in order, the investigation of which is the solution of Prop. IV., 



* Communicated by the Author. 



