PAPER BY PROF. HELMHOLTZ. 37 



equal to zero. Therefore, according to the well-knowu Green's theo- 

 rem,* 



where, on the left hand, the integration is to be extended over the whole 

 of the volume S, but on the right hand over the whole surface S whose 



elementary surface is designated by doj. If, now, '—^ is to be equal to 



zero for the whole surface, then the integral on the left hand must also 

 be zero, which can only be true when for the whole volume S 



that is to say, when there exists no motion whatever of the liquid. 

 Every motion within a simply connected space of a limited mass of 

 fluid that has a velocity potential is therefore necessarily connected 

 with a motion of the surface of the fluid. If this motion of the surface 



i. e., ^, is known completely, then the whole movement of the inclosed 



fluid mass is also thereby definitely determined. For suppose there are 

 two functions, cp, and (p,„ that simultaneously satisfy the equation 



in the interior of the space 8. and also the condition 



for the surface of 8. where ib indicates the value of — deduced from 



the assumed motion of the surface, then would the function {(p, — (p,,) 

 also satisfy the first condition for the interior of the space 8, but for 

 the surface this function would give 



Jn -"' 

 whence, as just shown, it would follow that for the whole interior of 8 

 we would have 



dx dy ?z 



Therefore both functions would also correspond to exactly the same 

 velocities throughout the whole interior of 8. 



Therefore rotations of liquid particles and circulatory motions within 

 simply-connected wholly inclosed spaces can only occur when no veloc- 

 ity potential exists. We can therefore in general characterize the mo- 

 tions in which a velocity potential does not exist, as vortex motions. 



*Thi8 is the proposition ia Crelle Journal, vol. liv, p. 108, already alluded to, and 

 which does uot bold good for complex or manifold-connected space. 



