APPLICABLE TO THE MOTION OF FLUIDS. 373 



First, it will appear by the following reasoning that 



udx + vdy + wdz = 0, 



is the differential equation of a surface which at a given instance cuts 

 at right angle the directions of motion of the particles through which 

 it passes*. Let qPp (in the figure) represent such 

 a surface, which, for brevity, we will call a surface 

 of displacement. Let P be a point in it, the co- 

 ordinates of which are x, y, z, and let R be any 

 other point indefinitely near, whose co-ordinates are 

 x + dx, y + dy, z + dz. Draw PQ in the direction 

 of the motion at P, and therefore perpendicular to the surface qPp, 

 and draw RQ perpendicular to PQ. Let PR = ds, and PQ = dr. Also 

 let PR make the angles a, fi, y with the axes of co-ordinates, PQ 

 make the angles a, fi', y with the same axes, and PR make the angle 

 with PQ. Then if V be the velocity at P, 



U — V COS a, V = V COS fi', w = V cos y, 



also, dx = dn cos a, dy = ds cos /3, dz = ds cos y. 



Hence, 



udx + vdy f wdz = Vds (cos a cos d + cos /3 cos /3' + cos 7 cos 7') 



= Vds cos 9 = Vdr. 



Now if the variation of the co-ordinates be from P to a point p indefi- 

 nitely near on the surface of displacement, dr = 0, and therefore, since 

 V does not vanish, 



udx + vdy + wdz = 0, 



which it was required to prove. 



Next, if r and r' be the principal radii of curvature at any point 

 of a surface, the differential equation of which is (d<p) = 0, it may be 

 shewn by the processes of Analytical Geometry, that, 



* See Mr. Earnshaw on Fluid Motion in the Cambridge Philosophical Transactions, 

 Vol. VI. Part 11. p. 204. 



Vol. VII. Part III. Ti 



