RECENT PROGRESS IN HYDRODYNAMICS. 61 



the space filled by the fltiicl are sought in terras of the time and position. 

 We have now to refer to a remarkable simplification introduced into the 

 equations of motion in the ' Lagrangian ' form, where the position of any 

 particle at any time is sought in terms of the time and its initial position. 

 This was effected by Weber ^ in 1868. Integrating the equation by 

 parts with respect to the time between the limits o and t, they take the 

 form 



dx dx dy dy . dz dz _ c7/\ 



dt da dt da dt da da ' 



where (x y z) is the position of the particle at time t, whose initial 

 position was {a h c), and whose initial velocities were ti^ Vq ivq ; where also 

 \ is given by 



-^ = Force potential — — - + ?> (vel)^. 

 dt ] p " 



I In the particular case where the initial velocities have a potential 

 Mq + tlX/da, . . . can be written dxfda, . . . , and the equations 

 with the equation of continuity give foar equations to determine the four 

 unknown functions x y z and x, of the first order but second degree. This^ 

 forms an easy proof of the theorem, once a velocity potential, always 

 one. 



We have already seen how Clebsch attacked the theory of the general 

 motion of a fluid, by employing three functions 0,m,\l/ where 



11 dd: + V dy + IV dz = df + m d\p, 



and it was pointed out that' m and \p by their intersections give vortex 

 lines. It is clear then that these surfaces must form two families which 

 contain the same particles of fluid at all times. The general problem of 

 taking as independent variables any system of surfaces always contain- 

 ing the same particles was taken up by Mr. Hill,^ apparently without a 

 knowledge of Clebsch's researches. He starts by taking any three in- 

 dependent sets of such surfaces (say P,Q,E,) which must bg three inde- 

 pendent solutions of the equation 



df , df , df , df ^ cf 

 at dx dy dz ct 



Transforming the equations to independent variables P.Q.R, and taking 

 the circulation round elementary circuits, he shows that 



udx + V dy + IV d~ = f^ dV + /, cZQ + f^dU + cZK 



where /i /2 /a are definite functions of P,Q,R. Further, the pressure is 

 given by 



£ +V -lU- + ^2 + ,,2\ = _ f]K 



P '\ J dt 



After proving that tbe vortex lines are the intersections of two surfaces 

 which satisfy cfjlt = 0, Q and R, are chosen to be such. This introduces a 

 considerable simplification and enables us to write the former equation in 

 the form ^^ dx + v dy -{■ w dz ^= dx + ^d-1/ where IXjU = 0,mH = 0. The 

 velocities are now in the form given by Clebsch, but the expression for 

 the pressure appears at first sight difierent. This is not so in realitv, as 



' ' Ueber eine Transformation der hj-drodynamischen Gleichungen,' Crelli; Ixviii. 

 = Lamb's treatise On the Motion of a Fluid, p. 18. 



' 'On some properties of the equations of hydrodynamics,' Quart. Jour, of Math. 

 Feb. 1880 ; vol. xvii. 



