72 PROFESSOR TAIT ON GREEN'S AND OTHER ALLIED THEOREMS, 



and the two latter equations give 



^/•S.VPVP^s = -fff^Wch +ff? 1 S.VYVvd8 , 

 = -fffPV 2 ~Pid<s +f/PS.VPJJvds , 



which are the common forms of Green's Theorem. Sir W. Thomson's extension 

 of it follows at once from the same proof. 



6. If P x be a many-valued function, but VPi single-valued, and if 2 be a 

 multiply-connected* space, the above expressions require a modification which 

 was first shown to be necessary by Helmholtz, and first supplied by Thomson. 

 For simplicity, suppose 2 to be doubly-connected (as a ring or endless rod, 

 whether knotted or not). Then if it be cut through by a surface s, it will be- 

 come simply-connected, but the surface-integrals have to be increased by terms 

 depending upon the portions thus added to the whole surface. In the first form 

 of Green's Theorem, just given, the only term altered is the last : and it is 

 obvious that if jh be the increase of P x after a complete circuit of the ring, the 

 portion to be added to the right hand side of the equation is 



p 1 ffS.VPTJvd8 



taken over the cutting surface only. Similar modifications are easily seen to be 

 produced by each additional complexity in the space 2. 



7. The immediate consequences of Green's theorem are well known, so that 

 I take only one instance. 



Let P and P x be the potentials of one and the same distribution of matter, 

 and let none of it be within 2. Then we have 



///{VVfck =ff¥S.VY\Jvds, 



so that if VP is zero all over the surface of 2, it is zero all through the interior, 

 i.e., the potential is constant inside 2. If P be the velocity-potential in the 

 irrotational motion of an incompressible fluid, this equation shows that there 

 can be no such motion of the fluid unless there is a normal motion at some part 

 of the bounding surface, so long at least as 2 is simply-connected. 

 Again, if 2 is an equipotential surface, 



JffiVPyds = FffS.VFTJvds = F/ffV 2 Fd? 



by the fundamental theorem. But there is by hypothesis no matter inside 2, 

 so this shows that the potential is constant throughout the interior. Thus there 

 can be no equipotential surface, not including some of the attracting matter, 

 within which the potential can change. Thus it cannot have a maximum or 

 minimum value at points unoccupied by matter. 



* Called by Helmholtz, after Ribmann, mehrfach zusammenhangend. In translating Helm- 

 holtz' s paper (Phil. Mag. 1867) I used the above as an English equivalent. Sir W. Thomson in his 

 great paper on Vortex Motion (Trans. R.S.E. 1868) uses the expression "multiply-continuous." 



