FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 105 



particle in space, an extension or contraction along three directions at 

 right angles, and, finally, a rotation about a temporary axis. The exist- 

 ence of a rotation is, however, excluded when a velocity potential 

 exists. Helmholtz designated forms of displacement, for which a 

 velocity potential is not to be derived, as vortex motions. In deter- 

 mining the change in the velocity of rotation during the continuance 

 of motion, he discovered that those x>articles of fluid which do not 

 already possess rotation do not have such motions imparted to them in 

 the progress of the disturbance. Defining a vortex line as a line whose 

 direction is throughout coincident with the direction of the instantane- 

 ously existing axes of rotations of the particles along it, a remarkable 

 law was deduced, which may be expressed as follows: A vortex line 

 remains continually with the same particles, progressing with these 

 particles through the fluid, and the value of the resulting velocity of 

 rotation for any particle of the fluid varies directly with the distance 

 of this particle from its neighbors in the vortex line. If, further, we 

 designate as a vortex thread the portion of the fluid inclosed within 

 an indefinitely thin mantle of vortex lines, the product of the velocity 

 of rotation by the cross section of the vortex thread is constant through- 

 out its whole length, and so remains during a progressive motion of 

 the same. It follows that a vortex thread can never cease within the 

 boundaries of the fluid, but must be either a closed riug wholly within 

 the fluid or must continue to its boundaries. In the attempt to deter- 

 mine from the velocity of rotation the velocity of translation, Helm- 

 holtz succeeded by methods of great mathematical interest in making 

 it possible to form a conception of the forms of motion, though the 

 complete analytical solution of the problem was possible only in the 

 simplest cases. Under certain preliminary circumstances relative to 

 the nature of the surroundings, the vortex threads and vortex rings 

 retain unchanged the same quantity of the fluid and are permanent. 

 In this case it was shown that two vortex rings whose axes are the 

 same and which have the same direction of rotation would proceed in 

 the same direction, the foremost becoming distended and moving slower 

 and slower, while the follower would concentrate itself and move faster, 

 till finally — provided the velocities of translation lie within certain lim- 

 its — the follower would overtake the leader, pass through it, and assume 

 the role which the other played before. This procedure would, how- 

 ever, in the actual phenomena of vortex motions be very soon inter- 

 rupted, owing to the friction. 



The regularity and relative stability of vortex phenomena have led 

 W. Thomson (Lord Kelvin) to put forward an interesting hypothesis 

 in which the atoms are supposed to have the form of vortex rings, with 

 the aim of uniting the theory of the continuity of matter and the 

 atomic theory. He has also succeeded in connecting the results of the 

 theory of vortex-atoms with the motion of solid bodies in fluids, and 

 thus to join with those investigations which aim to eliminate action at 



