306 Prof. Potter on Hydrodynamics. 



motion in a fluid, I now proceed to discuss the equations for the 

 general case, and maintain their accuracy on the following grounds 

 in adthtion to the theoretical reasons given above : — first, the ordi- 

 nary expressions which are deduced from the old theoiy, and 

 admitted as accurate, arise on proper considerations from my 

 equations ; and secondly, the properties of diverging streams of 

 fluid and of \-ibratory motion, which could not be deduced from 

 them, arise almost as simply as the others from my general 

 formula;. 



It is almost with regi'et that one sees the beautiful deductions 

 fi'om the equation of continuity to be imnecessary ; and I had 

 formerly thought that the ordinaiy equations required only the 

 substitution of an equation of form in place of the equation of con- 

 tinuity, which would have expressed the tendency in the atoms 

 to resume the cubical arrangement after a displacement, iu virtue 

 of their mutual repulsions. No such equation would, however, 

 render the theoiy correct in passing fi-om the ordinary equations 

 of equilibrimn to those of motion. As far as the equilibrium of 

 fluids is concerned, there is no need to refer to their intimate con- 

 stitution, although with more complexity the same results might 

 be obtained by considering the mutual pressures of the atoms. 



In hydi'odynamics, however, we must commence with the cir- 

 cumstances of the motion of the most elementaiy portion which 

 is capable of separate motion, or with those of an ultimate atom. 

 The mathematical expression for fluids iu a state of equilibriimi 

 not being based on atomic considerations, the equations of fluid 

 motion cannot be derived from them ; but an especial appUcation 

 of D'Alembert's principle must be made, and then the principle 

 of virtual velocities becomes available. 



Proposition. To form the general equations of motion for an 

 atom in a fluid. 



Let X, y, z be the coordinates of x 



an atom M, whose mass is m. 



X, y, Z the accelerating forces 

 acting upon it parallel to the 

 axes respectively. 



u, V, w the velocities of M par- 

 allel to the axes respectively. 



Let p be the pressm-e on a imit 

 of ai-ea, if as at ]\I, and Zp the dif- 

 ference of the pressui'cs from the 

 neighbom-ing atoms in the direc- 

 tion of motion, so that the moving force on the atom from this 

 cause is 



Zp X area of the face of the attributed cube of the atom. 



Let s be an arc of the path of M, and doc, dy, dz the compo- 

 nents of ds in the axes respectively. 



M 



