38-39] FLUX ACROSS A SPHERICAL SURFACE. 43 



with it by gradual variation of the radius, without ever passing 

 out of the region occupied by the irrotationally moving liquid. 

 We may therefore suppose the sphere contracted to a point, and so 

 obtain a simple proof of the theorem, first given by Gauss in his 

 memoir* on the theory of Attractions, that the mean value of < 

 over any spherical surface throughout the interior of which (1) 

 is satisfied, is equal to its value at the centre. 



The theorem, proved in Art. 38, that </> cannot be a maximum 

 or a minimum at a point in the interior of the fluid, is an obvious 

 consequence of the above. 



The above proof appears to be due, in principle, to Frost f. Another 

 demonstration, somewhat different in form, has been given by Lord Eayleigh|. 

 The equation (1), being linear, will be satisfied by the arithmetic mean of any 

 number of separate solutions < 1} < 2 , < 3 ,.... Let us suppose an infinite number 

 of systems of rectangular axes to be arranged uniformly about any point P as 

 origin, and let < 15 $ 2 , $ 3) ... be the velocity-potentials of motions which are 

 the same with respect to these systems as the original motion < is with 

 respect to the system x, y, z. In this case the arithmetic mean ($, say) of the 

 functions <f> lt $ 2 , </> 3) ... will be a function of r, the distance from P, only. 

 Expressing that in the motion (if any) represented by $, the flux across any 

 spherical surface which can be contracted to a point, without passing out of 

 the region occupied by the fluid, would be zero, we have 



or = const. 



Again, let us suppose that the region occupied by the irrota- 

 tionally moving fluid is 'periphractic,' i.e. that it is limited 

 internally by one or more closed surfaces, and let us apply (2) to 

 the space included between one (or more) of these internal 

 boundaries, and a spherical surface completely enclosing it and 

 lying wholly in the fluid. If 4-Trif denote the total flux into 

 this region, across the internal boundary, we find, with the 

 same notation as before, 



JJ dr 



* " Allgemeine Lehrsiitze, u. s. \v.," Eesultate aus den Beobachtungen des mag- 

 netischen Vereins, 1839 ; Werke, Gottingen, 187080, t. v., p. 199. 



t Quarterly Journal of Mathematics, t. xii. (1873). 



J Messenger of Mathematics, t. vii., p. 69 (1878). 



See Maxwell, Electricity and Magnetism, Arts. 18, 22. A region is said to be 

 * aperiphractic ' when every closed surface drawn in it can be contracted to a point 

 without passing out of the region. 



