98 



In other cases there is no direct metliod of procedure. The inverse 

 process of finding what boundary conditions Avill give Ij^uown solutions 

 of Laplace's equation is used. Avitli the hope uf 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 

 velocity-potential which satisties the equation 



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

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

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

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

 is "steaciy" or not arid true, therefore, when the extraneous force is 

 gravity. 



Much woi'k has been done on the motion of many of the regular solids 

 immersed in a liquid, when acted upon liy a system of impulsive forces 

 and also by constant forces. The motions of the liquid in the neighl)or- 

 hood of such solids has also been discussed. Both tidal Avaves 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 neighlioring bodies of water has been worked out by Professor 

 R. S. Woodward and others. 



Perhaps the most familiar problem of the effect of an extraneous 

 force upon a body of liquid, is the "Torricelli Theorem" on the efflux of a 

 liquid fi'om 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 --- ^0, -^— = and -^^^ = 0; and giving the well-known re- 

 dt dt dt 



suit q''^ = 2 gz, where q is the velocity. In case the liquid rotates under 



the influence of gravity angular velocity is introduced, giving ~z — = 2w. 



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- 

 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 



