99 



belong to ortliogoual systems of eiu-A-es and no method has yet lieen de- 

 vised by means of which tlie lines could he drawn under specified condi- 

 tions. 



It was lioped that some graphical method applicable to all cases 

 might be found in connection with the present work, Init thus far none 

 has been discovered that is at all general. I have found the eciuipotential 

 lines and lines of flow for a rectangular area where a constant extrane- 

 ous force is acting. 



Taking the liquid as incompressible since the external forces is con- 

 stant the motion is steady and the velocity potential may be made to 

 satisfy the equation 



^ (Vy _ 

 dx^'^cly2 — " 



and — ^ = ku, -^ = kw. 



t5x f^z 



A constant must he added to one of these velocities to express the effect 

 of the constant f<jrce. This is more clearly seen perhaps in the case of 

 vertical motions due to the force of gravity. In this case the constant to 

 be added to w is of course g and since this is a constant Laplace's equa- 

 tion is still satisfied. The lines of flow and equipotential lines are no 

 longer orthogonal, but are, as we shall preseutlj- see, inclined at different 

 angles, lieing tangent at some points of the interior. 



If the area be taken in the sphere of attraction of the earth and near 

 enough so that the attraction may be taken as constant we shall have 



Uzrik -L- 



dx 



V = k '— + kpg. 

 dz^ ^ 



where satisfies Laplace's equation. 



Professor C. 8. Slichter^ has shown that the motions in an area 

 A B C D, Fig. 1, filled with sand and having water flowing through it, 

 entering along A B and flowing out along A D— the sides B C and C D 

 being impervious— may be fully discussed by replacing the sand and water 

 by a perfect liquid having a velocity potential, and that the velocity po- 

 tential in this case would be identical with the pressure function. This 

 being true, it is possible to find the pressure at any point in the interior 

 as well as the component velocities at these points, just as soon as the 



1. 19th Annual Report, U. S. Geological Survey, Part II. 



