

PAPER BY PROF. HELMHOLTZ. 37 



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

 rem,* 



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

 of the volume 8, but on the right hand over the whole surface 8 whose 



elementary surface is designated by dco. If, now f ^ 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 8 



dcp _ d<p__ dq> _ 

 dx ?y d*~ ' 



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. <?., ^, is known completely, then the whole movement of the inclosed 



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

 two functions, q>, and <p in that simultaneously satisfy the equation 



pq> fcp ? 2 qp _ 

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



for the surface of 8, where ip indicates the value of —-- deduced from 



the assumed motion of the surface, then would the function ((Pz — cp,,) 

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

 the surface this function would give 



3{<P,— < Pi/) == q. 



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

 we would have 



H<P-/—<P n) — H<P- <Pn) _ H<P, — <P„) _ q 



doc dy dz 



" 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. 



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

 which does not bold good for complex or manifold-connected space. 



