92-93] SPHERE WITH CONCENTRIC BOUNDARY. 133 



As b diminishes from oo to a, this increases continually from 

 to GO , in accordance with Lord Kelvin's minimum theorem (Art. 45). 

 In other words, the introduction of a rigid spherical partition in an 

 infinite mass of liquid acts as a constraint increasing the kinetic 

 energy for a given velocity, and so virtually increasing the inertia 

 of the system. 



93. In all cases where the motion of a liquid takes place in a 

 series of planes passing through a common line, and is the same in 

 each such plane, there exists a stream-function analogous in some 

 of its properties to the two-dimensional stream-function of the 

 last Chapter. If in any plane through the axis of symmetry we 

 take two points A and P, of which A is arbitrary, but fixed, while 

 P is variable, then considering the annular surface generated by 

 any line AP, it is plain that the flux across this surface is a 

 function of the position of P. Denoting this function by 2-Tn/r, 

 and taking the axis of x to coincide with that of symmetry, we 

 may say that ty is a function of x and vr, where x is the abscissa of 

 P, and CT, = (y 2 + z 2 )*, is its distance from the axis. The curves 

 ijr = const, are evidently stream-lines. 



If P' be a point infinitely near to P in a meridian plane, it 

 follows from the above definition that the velocity normal to PP' 

 is equal to 



27TOT.PP" 



whence, taking PP' parallel first to or and then to x, 



1 d^ 1 d^ 



u= --- ^- , u = - f- .................. (1), 



& div OT dx 



where u and u are the components of fluid velocity in the directions 

 of x and OT respectively, the convention as to sign being similar to 

 that of Art. 59. 



These kinematical relations may also be inferred from the 

 form which the equation of continuity takes under the present 

 circumstances. If we express that the total flux into the annular 

 space generated by the revolution of an elementary rectangle 

 is zero, we find 



(V . 27TW&C) &T = 0, 



. 

 CiX Cttzr 



