Prof. Sylvester ofi the Motion and Rest of Fluids. 453 



Now by drawing through all the points in a plane parallel 

 to X y^ planes parallel to y x, we may cover the whole surface ; 

 hence 4> is constant all over the surface bounding the fluid. 



.-. ^^ dx + Y, dy + Z^dz — for all va-\ , . 

 nations of c? a?, dy, dz taken upon the surface . .j "* ^ '' 



The equations 1, 2, 3, 4, 5, 6 are coincident with those ob- 

 tained by the usual method ; with this diiFerence, that X^ Y^ Z^ 



here take the place of j— —- -r— . 



dx dy dz 



Thus then we have obtained all the conditions requisite 

 for determining the motion of fluids from the universal princi- 

 ple of least constraint conjoined with the specific character of 

 the system in question. 



General RemarJcs. 



In the case of equilibrium, i. e. in the case where no par- 

 ticle moves, we have X^ + X = 0, Y^ + Y = 0, Zy + Z = 0. 

 Hence ^dx+Ydy+7jdz is a complete differential al- 

 ways and zero for the surface. 



The above results have been obtained upon the principles 

 of the differential calculus, and the continuity of the forces 

 has been tacitly assumed. If now we were to suppose forces 

 of finite magnitude (as compared with the *mhole sum acting 

 upon the entire system) to be applied to a layer of single par- 

 ticles or to a layer of a thickness of the same order of magni- 

 tude as the distances between the particles themselves, (which 

 has been treated as an infinitesimal) itVould appear that our- 

 results would be no longer applicable, just in the same manner 

 as it would be erroneous to apply the principle of vis-viva 

 (for example) without modification, to the case of impulsive 

 forces, because we had deduced it by the calculus in the case 

 of the motion being continuous. Hence the above equations 

 ought not strictly to apply to the motion or rest of a fluid 

 contained between physical surfaces ; for the pressure afforded 

 by these surfaces, whatever its actual value may be, we know 

 a priori is commensurable with the whole amount of force 

 acting on the fluid; but the immediate application of this 

 pressure (alias repulsive force) is confined to the bounding 

 layer of fluid particles, or at most extends to a- distance bearing 

 a low ratio to the distances between the particles themselves. 



According to the non-applicability of the equations for 

 free fluids to the case of fluids confined at the boundaries and 

 to an independent investigation upon the minimum principle 

 for this class of problems, it is that I look for the true expla- 

 nation of the phaenomena of capillary attraction (vulgarly so 

 called). 



University College, Oct. 25, 1838. J. J. SyLVE^TSR. 



