41-42] REGION EXTENDING TO INFINITY. 47 



in the mean values can arise from this cause. On the other hand, 

 when Q, Q' fall without 2, the corresponding values of <f> cannot 

 differ by so much as e . QQ', for e is by definition a superior limit 

 to the rate of variation of </>. Hence, the mean values of $ over 

 the two spherical surfaces must differ by less than e . QQ'. Since 

 QQ' is finite, whilst e may by taking 2 large enough be made as 

 small as we please, the difference of the mean values may, by 

 taking P sufficiently distant, be made infinitely small. 



Now we have seen in Art. 39, that the mean value of < over 

 the inner sphere is equal to its value at P, and that the mean 

 value over the outer sphere is (since M = 0) equal to a constant 

 quantity G. Hence, ultimately, the value of < at infinity tends 

 everywhere to the constant value C. 



The same result holds even if the normal velocity be not 

 zero over the internal boundary ; for in the theorem of Art. 39 

 M is divided by r, which is in our case infinite. 



It follows that if d(f>/dn at all points of the internal 

 boundary, and if the fluid be at rest at infinity, it must be every- 

 where at rest. For no lines of motion can begin or end on the 

 internal boundary. Hence such lines, if they existed, must come 

 from an infinite distance, traverse the region occupied by the 

 fluid, and pass off again to infinity ; i.e. they must form infinitely 

 long courses between places where < has, within an infinitely 

 small amount, the same value C, which is impossible. 



The theorem that, if the fluid be at rest at infinity, the motion 

 is determinate when the value of dcfr/dn is given over the in- 

 ternal boundary, follows by the same argument as in Art. 40. 



Green's Theorem. 



42. In treatises on Electrostatics, &c., many important pro- 

 perties of the potential are usually proved by means of a certain 

 theorem due to Green. Of these the most important from our 

 present point of view have already been given ; but as the 

 theorem in question leads, amongst other things, to a useful 

 expression for the kinetic energy in any case of irrotational 

 motion, some account of it will properly find a place here. 



Let U, V, W be any three functions which are finite, con- 

 tinuous, and single-valued at all points of a connected region 



