Manchester Memoirs, V\d. Ivi. (191 2), No. 10. 



The particular integral solution. 



Assume 



and 





r, dv dw 



ay dz 



^ du , div 



^' d.v dz 



r, du , dv 



^.T dy 



f^ _ dw dv 

 ^'~~dy~^' 



'T' _ _du _ dtv 



TT dv du , V 



Ui=-~-~-^, (3). 



Then we shall require that 



and 



^0j _ d(f) d(p„ _ d(p d(ps _ dif) . 

 dx dx dy dy dz dz 



div dv r du dw ,, dv du -.r , •, 



dy dz dz dx dx dy 



as the conditions for satisfying (i) and (2). 



I have retained </>i , r/)„, ^3, with their definition in (4), 

 because, although we might replace each of them by (p. 

 The single function (p is ordinarily introduced, partly for 

 the sake of form and partly because in ordinary dynamical 

 problems it represents a form of "energy," but in the 

 case under consideration these arguments are void, and 

 each of the quantities 0,, 0_, , </>. will have its simplest 

 value. For an illustration, note a heavy body rotating 



