178 Mr CHALLIS's RESEARCHES IN THE THEORY 



6. If we put (f} = (pi + (p2 + (f>3 + Sac. we shall have 



d^ d^_ (d^ d^\ Id^ dy,\ id^ d-^<pA , . _ „ 

 daf "*" df ~ \dx' ^ dfj '^ \dx' "^ df) ■*■ \dx' ^ df)'^^-~^- 



Hence if there be any number of functions which severally satisfy the 

 given equation, the sum of these will satisfy it. But from what has 

 been proved above, if 



<p\, 02. 03, &c. will severally satisfy it; therefore 0i + 02 1- 0, + &c. 

 will also. And we have, 



dx dx dx 



v=^ +^ +^ +&,c 

 dy dy dy 



These equations prove that the velocity at any point may be the re- 

 sultant of several velocities produced by different causes; and that any 

 given cause will have the same effect in producing velocity at a given 

 point, whether or not other causes may be operating to produce 

 velocities at the same point. 



7. We may here also determine the manner in which the motion 

 of the fluid is affected, when the rectilinear transmission of an impulse 

 tending from any centre is interrupted by a plane surface. For suppose 

 two impulses tending from two centres to be of equal magnitude and in 

 every respect alike. Then if the straight line joining these centres be 

 bisected at right angles by a plane, there will be no motion of the par- 

 ticles contiguous to the plane in a direction perpendicular to it, because 

 the resultant of the velocities from the two causes must lie wholly in 



