THE 



LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



SUPPLEMENT to VOL. XXXIII. FOURTH SERIES. 



LXIII. On Integrals of the Hydro dynamical Equations, which 

 express Vortex-motion. By H. Helmholtz*. 



HITHERTO, in integrating the hydrodynamical equations, 

 the assumption has been made that the components of 

 the velocity of each element of the fluid in three directions at 

 right angles to each other are the differential coefficients, with 

 reference to the coordinates, of a definite function which we shall 

 call the velocity-potential. Lagrangefno doubt has shown that 

 this assumption is lawful if the motion of the fluid has been pro- 

 duced by, and continued under, the action of forces which have a 

 potential ; and also that the influence of moving solids which are 

 in contact with the fluid does not affect the lawfulness of the 

 assumption, xlnd, since the greater number of natural forces 

 which can be defined with mathematical strictness can be ex- 

 pressed as differential coefficients of a potential, by far the greater 

 number of mathematically investigable cases of fluid-motion 

 belong to the class in which a velocity-potential exists. 



Yet Euler % has distinctly pointed out that there are cases of 

 fluid-motion in which no velocity-potential exists, — for instance, 

 the rotation of a fluid about an axis when every element has the 

 same angular velocity. Among the forces which can produce 

 such motions may be named magnetic attractions acting upon a 

 fluid conducting electric currents, and particularly friction, 

 whether among the elements of the fluid or against fixed bodies. 

 The effect of fluid friction has not hitherto been mathematically 

 defined ; yet it is very great, and, except in the case of indefi- 

 nitely small oscillations, produces most marked differences be- 

 tween theory and fact. The difficulty of defining this effect, 

 and of finding expressions for its measurement, mainly con- 



* From Crelle's Journal, vol. lv. (1858), kindly communicated by Pro- 

 fessor Tait. 



f Mecanique Analytique (Paris, 1815), vol. ii. p. 304. 



% Histoire de V Academie des Sciences de Berlin (1755), p. 292. 

 Phil. Mag. S. 4. No. 2.26. Suppl. Vol. 33. 2 K 



