68 REPOKT— 1881. 



continuity, and they are thus led to introduce -what they cull differe'rlt 

 orders of motion. The functions produced after n such operations give 

 the n th order of motion. If the functions of the n th order are expressible 

 by means of a scalar potential, then the functions of a higher order do not 

 occur. Looked at from the point of view as an investigation of the dis- 

 tribution of vorticity in a fluid motion whose vector-potential is given, it 

 may be possibly of some value, but I cannot help thinking that both 

 Prof. Rowland and Dr. Craig have somewhat overvalued its importance.' 



(c) Discontinuous Motion. Jets. 



In a fluid in motion the pressure is, in general, given as a 

 continuous function of position, and may, therefore, so far as the 

 analytical treatment is concerned, become zero or negative. But, in 

 this case, clearly, the fluid will cease to be continuous, and free 

 surfaces will be set up inside the fluid,^ or surfaces on the two sides of 

 which the tangential motions are different. It is curious that so evident 

 a fact as this seems not to have been noticed by mathematicians in general 

 until it was pointed out by Helmholtz,^ in 1868, though Stokes"* had drawn 

 attention to it twenty years before. He had already, in his before- 

 inentioned paper on vortex motion, considered, in passing, the case where 

 the tangential motions are diSerent on the two sides of a surface, and 

 had shown how the motions ought to be represented mathematically, by 

 supposing the surface a continuous vortex-sheet, the vortex-axis at any 

 point being parallel to the resultant of the two tangential velocities. In 

 the later paper he points out that wherever fluid is flowing across a sharp 

 edge, the velocity at the point, on the ordinary theory, would be infinite, 

 and the pressure negative infinity, consequently a surface of separation 

 be established. Tliis would also be the case with gases; but a 

 curious exception occurs, provided they obeyed Boyle's law alone, and 

 did not change their temperature on account of change of volume. 

 Here log p takes the place of p, and, clearly, log p may become negative 

 without necessitating a break of continuity in the fluid. This paper is 

 particularly important, as containing the first successful attempt at the 

 solution of a problem where discontinuity ensues. It is evident that 

 along any surface of discontinuity the pressures on both sides must be 

 the same ; and if the fluid on one side be at rest, under the action of no 



' The results are most easily proved and their meaning most clearly seen by the 

 quaternion method. If ^ be the quaternion potential, ff the velocity, then ( Q. Jour. 

 Math. xiv. p. 284), 



V^S5' = Svo' = a = y q «=po 

 2p = V ff = Pi whence S vpi = 

 and p„ =Vvp„-i=V''o- , S vp„ =0 



Also if p„ is derivable from a scalar potential v Pn = 0, or all successive orders 

 must vanisli. The onl}- question now of any interest seems to me to be the investi- 

 gation of the effect on the nature of the motions when the ■«"' operation gives zero 

 result ; but whether this is important remains doubtful. 



- Thus consider a sphere in motion, in an infinite non-gravitating fluid, whose 

 surface is under a constant pressure. The fluid will move in the well-known manner, 

 a nd t he sphere with constant velocity, provided the velocity be not greater than 

 '/2j)jp. If it is greater, there will be a hollow formed in the rear until the sphere 

 has been reduced to this limiting velocity. 



^ ' Ueber discontinuirliche Fliissigkeitsbewegungen,' Monatib. der Ic. Akad. Berl., 

 1868, p. 215, translated in the Phil. ''Mag. (4), 36, p. 337. 



* ' On the Critical Values of the Sums of Periodic Series,' Cam. Phil. Soc. Tram., 

 viii. p. 533, or reprint, p. 310, 



