60 KKPORT— 1881. 



tion may be drawn here to the new method by which he expressed the 

 velocities of a fluid in the case of the most general motion by means of 

 functions L,M,]Sr which give the velocities from 



ti = cU'ldx + cm/dy - d^i/dz, &c. 



The L,M,N are in reality the tensors of the three rectangular com- 

 ponents of a vector, now called a vector potential ; in fact, the velocities 

 are here expressed by a quaternion potential. Calling the potential q, the 



For vortex rings symmetrical about an axis, Helmlioltz transforms the 

 vector potential into another, which is nothing else than Stokes' stream 

 function for such motions divided by the distance from the axis. 



The next great advance in theory was due to the piiblication in 1867 

 of Thomson and Tait's 'Natural Philosophy.' Hei'e, for the first time, 

 Lagrange's equations of motion were applied, though without any direct 

 pi'oof that the equations were applicable to cases in which there is an 

 infinite degree of freedom and in which a portion of the generalised co- 

 ordinates do not ajjpear. Objections were raised by several mathema- 

 ticians to this application, amongst whom may be mentioned Purser ^ and 

 Boltzmann.^ As a matter of fact the equations are only directly applicable 

 under the conditions mentioned below. In the second edition, published in 

 1879, this question was considered under a general theory of ' ignoration of 

 co-ordinates.' Starting with the expression for the energy containing all 

 the generalised velocities, the generalised equations for the co-ordinates 

 ignored are written down and integrated once on the supposition that the 

 force components corresponding to the ignored co-ordinates do not occur. 

 These give a number of equations, containing the velocities, equal in 

 number to the ignored co-ordinates, and of the form — linear function of 

 the velocities := constant. By means of these, therefore, the ignored 

 velocities can be eliminated from the expression for the energy. This is 

 done, and Lagrange's equations for the non-ignored co-ordinates are 

 transformed to apply to the energy as expressed in the new form. 

 Naturally this is more complicated than the ordinary Lagrangian form, 

 but when we have to do with fluid motion, where the motion commences 

 from rest, or can be brought to rest without application of force com- 

 ponents corresponding to the ignored co-ordinates, the constants intro- 

 duced in the first integration are all zero. "When this is the case the 

 transformed Lagrangian equations reduce to the ordinaiy form. The 

 half-dozen pages devoted to fluid motion in the first volume have altogether 

 transformed the methods of hydrodynamics, and possibly have been a 

 cause why the papers of Clebsch already referred to have been allowed to 

 fall into neglect.** 



All the foregoing investigations have pi'oceeded on the method of the 

 so-called Euler's equations, i.e. where the velocities at different points of 



' Quarterly Journal, xiv. p. 284. 



- ' On the applicability oO Lagrange's equations in certain cases of fluid motion,' 

 Phil. Mafl. (5), vi. p. 354. 



' ' Ueber die Druckkriifte, welche auf Binge wirksam sind, die in bewegte 

 Fliissigkeit tauchen,' Borch. Ixxiii. p. 111. 



■* For further notices on the equation of motion see p. 16, below. 



