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LXIII. On the Motion and Best of Fluids. By J. J. Syl- 

 vester, Professor of 'Natural Philosophy in University CoU 

 lege, London. 



l\/r OSTROGRADSKY's memoir on this subject inserted 

 -L'-I- o in the Scientific Memoirs seems to have excited much 

 attention, and has been made the occasion of some annotations 

 by a distinguished writer in the Philosophical Magazine. Mr. 

 Ivory's recent papers in the same periodical must still more 

 tend to invest with a new interest all such speculations. It 

 seems to me desirable therefore to present the theory of fluids 

 in all the simplicity of which it is susceptible. 



I consider a fluid as a collection of particles subject to 

 some law of relative position other than that of rigidity. 

 These particles by their mutual actions maintain the con- 

 nexions of the system. As to the law of force between them 

 we know nothing; but I assume it is a general principle of 

 nature, that for each instant of time the sum of the internal ac- 

 tions (reckoned by the product of each particle into the square 

 of the space due to the internal force acting on it) is a minimum. 

 This in fact is Gauss's principle of least restraint. We may 

 if we please split this principle into two parts ; that is to say, 

 assume that the internal system of forces is always such as if 

 acting alone would keep the fluid at rest; and then again 

 assume that any equilibriating system of forces must be subject 

 to the law of virtual velocities. I say assume, because it is im- 

 possible a priori to prove this. 



Lagrange's so-called demonstration is unworthy of his name, 

 and (albeit sanctioned by the powerful oral authority of an 

 ex-Cambridge Professor) contrary alike to sense and honesty. 

 It is better therefore at once to proceed upon Gauss's principle. 

 It might easily be shown that this is in effect tantamount 

 in all cases to D'Alembert's and Lagrange's principles com- 

 bined. 



Before entering upon the investigation I may call attention 



to one point of great analytical interest, and relating to the 



difficult subject of the algebraical sign, viz. that if the density 



of a point {x, y) in any circumscribed space be expressed by the 



.^ du dv 1 , 

 quantity -t— + — — so that the mass is 



it tV €L 1i 



that is not equivalent to 

 Phil. Mag. S. 3. Vol. 13. No. 84-. 1838. 2 G 



