38 IRROTATIONAL MOTION. [CHAP. III 



the arrows, the positive side of the surface being that which faces 

 the reader. 



The value of the surface-integral taken over a dosed surface is 

 zero. 



It should be noticed that (5) is a theorem of pure mathe- 

 matics, and is true whatever functions u, v, w may be of x, y, 2, 

 provided only they be continuous over the surface*. 



34. The rest of this chapter is devoted to a study of the 

 kinematical properties of irrotational motion in general, as defined 

 by the equations 



= 0, 77 = 0, f = 0. 



The existence and properties of the velocity-potential in the 

 various cases that may arise will appear as consequences of this 

 definition. 



The physical importance of the subject rests on the fact that 

 if the motion of any portion of a fluid mass be irrotational at any 

 one instant it will under certain very general conditions continue 

 to be irrotational. Practically, as will be seen, this has already 

 been established by Lagrange's theorem, proved in Art. 18, but 

 the importance of the matter warrants a repetition of the investi- 

 gation, in the Eulerian notation, in the form originally given by 

 Lord Kelvin f. 



Consider first any terminated line AB drawn in the fluid, and 

 suppose every point of this line to move always with the velocity 

 of the fluid at that point. Let us calculate the rate at which the 

 flow along this line, from A to B, is increasing. If Sx, $y, z be 

 the projections on the co-ordinate axes of an element of the line, 



D Dux DBx 



we have Dt (u ^ = Dt S * + U ~W 



Now DSx/Dt, the rate at which Sx is increasing in consequence of 

 the motion of the fluid, is equal to the difference of the velocities 

 parallel to x at its two ends, i.e. to Bu ; and the value of D-u/Dt is 

 given in Art. 6. Hence, and by similar considerations, we find, if 

 p be a function of p only, and if the extraneous forces X, Y, Z 

 have a potential ft, 



y- (ux + vSy + w&z) - Sfl + uSu + vBv + wSw. 

 JJt p 



* It is not necessary that their differential coefficients should be continuous, 

 t I.e. ante p. 35. 



