202 DYNAMICAL THEORY OF SOUND 



The instantaneous configuration of the " equipotential sur- 

 faces" </> = const, indicates at once the distribution of velocity, 

 as regards both magnitude and direction. 

 Suppose two consecutive surfaces to be 

 drawn, for which the values of < differ by 

 S<. Let PP' be drawn normal to these, and 

 PP l parallel to x\ and let PP'=v. Ac- 

 cording to (3) the velocity at P, resolved Flg< 64 ' 



in the direction PP ly is 



3 PP' 8 



ultimately, if I denote the cosine of the angle which the normal 

 PP' makes with Ox. From this, and from the analogous forms 

 of v, w t it is seen that the velocity at P is normal to the equi- 

 potential surface passing through that point, and is equal in 

 magnitude to the limiting value of </>/*/. Hence if a system 

 of surfaces be drawn corresponding to values of <f> which differ 

 by equal infinitesimal amounts, the velocity is everywhere 

 orthogonal to these, and inversely proportional to Sv, the distance 

 between consecutive surfaces. More precisely, the velocity is 

 everywhere in the direction in which < decreases* most rapidly, 

 and is equal in absolute value to the gradient of <. 



If we draw a linear element PQ (= Bs) in any other direction, 

 the velocity resolved in the direction of PQ is equal to the limit of 



-. ............... (6) 



or - d(f>/ds. 



The cases in which a velocity-potential exists include all 

 those where, in the region considered, the fluid was initially at 

 rest, for we may then put </> = 0, simply, and the subsequent 

 value is 



4>=c 2 f sdt ........................ .(?) 



J o 



This will hold whenever the motion has been originated by the 

 vibration of solid or other bodies. 



* It should be mentioned that in many books is taken with the opposite 

 sign; thus u = d<J>jdx, &c. 



