204 Mb EARNSHAW, ON FLUID MOTION. 



GENERAL PROPERTY. 



1. If in a fluid meditim we describe a surface whose differential 



equation is 



udx + vdy + wd% = 0, 



the 7notion of each particle through ivhich this surface passes is in the di- 

 rection of the normal at the point where the particle is situated. 



u, V, w are the velocities, estimated parallel to the co-ordinate axes, 

 of the particle whose co-ordinates are x, y, %. Let there be another 

 particle in the surface very near to this; and let its co-ordinates be 

 X -V dx, y + dy, %-\-d%; and let ds be their distance from each other ; 

 by a, ji, 7 denote the inclinations of ds to the axes. Let also V be 

 the velocity of the former particle, and by a', /3', y denote the incli- 

 nations of the direction in which V takes place to the three axes. 



Then, = udx + vdy + wd% 



tu dx V dy w dx \ 

 = l^ds. [y--^ + y--^ + -p-^J 



= Vds . (cos a cos a + COS /3' COS (i + cos y' cos 7) : 



and since, from the nature of the case, neither y nor ds is equal to 

 zero, 



.•. cos a' cos a + COS /3' COS/3 + COS 7' COS 7 = 0. 



But the left hand member expresses the cosine of the inclination 

 of V to ds, which being equal to zero, V and ds must be at right 

 angles to each other; that is, the motion of the particle whose co- 

 ordinates are x, y, », takes place in a direction perpendicular to the 

 surface whose equation is 



udx + vdy + wdx = 0. 



We may simultaneously draw surfaces of this nature through all 

 parts of the fluid in motion, and shall thus obtain the direction of 

 the motion of every particle. It is manifest that these surfaces are 

 very analogous to the level-surfaces, which occur in investigations con- 

 cerning the equilibrium of heterogeneous fluids. 



