98 



In other cases there is no direct method of procediire. The inverse 

 process of finding whnt lionndary conditions will give known solutions 

 of Laplace's equation is used, with the hope of finding the desired solu- 

 tion. The method of images is also applicable to some cases, more 

 especially perhaps in the case of rotational motion. 



For the irrotational motion of a perfect liquid there always exists a 

 velocitj^-potential Mhicli satislies the equation 



V-'?> = 0. 

 The potential o and the rtctaugalar velocities u, v and w may be 

 found from the given conditio.is, for all points of the intericn-. The 

 potential being always least at the boundary the lines of flow and equi- 

 potential lines begin and end there. This is true whether the motion 

 is "steady" or nut and true, therefore, when the extrane;ins force is 

 gravity. 



Much work lias been doiu' on the motion of many of the regular solids 

 immersed in a liiiuld, when acled upon Ity a system of impulsive forces 

 and also by constant forces. The niotioiis of the liquid in Uw neighbor- 

 hood of such solids has also been discussed. Both tidal waves and waves 

 due to local causes have been investigated and their properties discussed 

 to some extent. The related problem of the effect of high land masses 

 upon neighboring bodies of water has been worked out liy Professor 

 R. S. Woodward and others. 



Perhaps the most familiar problem of the effect of an extraneous i 

 force upon a body of li(iuid, is the "Torricelli Theorem" on the efflux of a 

 liquid from an aperture in the side or bottom of the containing vessel. 

 There the vessel is kept filled to a constant level the motion becomes 



steady making ^'^^O- J^ = ^^^^^ = ^' '"'"^ ^'"'"'^ ^^'^ well-known re- 

 sult q2 = 2 gz, where q is the velocity. In case the liquid rotates under 



. . dv du „ I 



the influence of gravity angular velocity is introduced, givmg ^ — — — ^w, j 



Showing that a velocity potential does not exist, and that such motion 

 could not take place in a perfect liquid. , 



Cases of motion where no extraneous forces are acting have been com- j 

 pletely worked out by methods of conjugate functions and the theory of 

 images, iln these cases the lines of flow and equipotential lines are 

 orthogonal systems of curves, and methods of plotting such are easily 

 devised. But when extraneous forces are acting these lines no longer 



