534 Sir W. Thomson on Maximum and 



be called a vortex sponge is formed ; a mixture homogeneous* 

 on a large scale, but consisting of portions of rotational and 

 irrotational fluid, more and more finely mixed together as 

 time advances. The mixture is altogether analogous to the 

 mixture of the white and yellow of an egg whipped together 

 in the well-known culinary operation. Let V be the radius 

 of the cylindric vortex sponge, and ? r its mean molecular 

 rotation, which is the same in all sensibly large parts. 

 Then, b being as before the radius of the original vortex 

 column, we have 



T = £V, fromr=0 to r=¥, 

 and 



T = ?b /2 jr, from r = V to r = a ; 

 where 



and ti/t 



18. Once more, hold the cylindric case from going round 

 in space, and continue holding it until some more moment of 

 momentum is stopped from the fluid. Then leave it to itself 

 again. The vortex sponge will swell by the mingling with it 



* Note added May 13, 1887. — 1 have had some difficulty in now proving 

 these assertions (§§ 17 and 18) of 1880. Here is proof. Denoting for 

 brevity l/2ir of the moment of momentum by /x, and 1/2-ir of the energy 

 "by e, we have 



li =\ a Tr . r dr, and e = \ f a T 2 . r dr. 



The problem is to make e least possible, subject to the conditions : (1) that 

 ix has a given value ; (2) that 



= £, and>0; 



x/T dT\: 



2 \7 + dFJ' 

 and (3) that when r = a, T = (b 2 /a ; this last condition being the resultant of 



which expresses that the total vorticity is equal to that of £ uniform within 

 the radius b. The configuration described in the last three sentences of 

 § 17 and the first three of § 18 clearly solve the problem when 



M<:±7r#>V-& 2 ); or /x>K&V. 

 The fourth sentence of § 18 solves it when 



The second paragraph of § 18 solves it when 



M > 1 7r£bXa 2 -b 2 ); or h < £ £6 V. 



