that B becomes identical with -for the flux of force w (§ 83).

This extremely important law was established by HELMHOLTZ, as a consequence of the theorem that the rotational velocity increases in proportion to the distance between two neighbouring particles of the same vortex line.

Since any given portion of a vortex filament always consists of identically the same liquid elements, and owing to the incompressibility of the liquid, the volume of this portion cannot change, the angular velocity a must vary inversely as the cross-section; that is the product of the two must be constant.

Thus a vortex field is a complete mechanical analogue of a field of force.

As in the case of the latter the field may be supposed to be partitioned into tubes of force, along each of which the flux of force remains constant, so in the case of the former we have the constancy of vortex strength along each vortex filament.

The more rigorous mathematical analysis given in the second part of the book will enable us to follow the analogy further.

135. Experiments with smoke rings. Smoke rings, such as a skilful smoker blows with his mouth, show very clearly the motion corresponding to a tube of force which runs. completely round a closed curve, such as a circle. Tubes of force of this kind will have to be more particularly considered in Section II.

Smoke rings may be produced in greater numbers and with greater regularity by making use of a cylindrical tin or cardboard case placed with its axis horizontal, and having a circular hole pierced through the middle of its base. Smoke is blown in through the other end of the case, which is then closed by a lid. When the air within the case has come to rest, a succession of blows is struck upon the lid, and at each blow a well-formed ring issues from the circular hole in the base. The admixture of smoke particles enables us to follow and recognise the movements of the air expelled from the box, and it is always with this particular portion of air that the vortex motion is associated.

The rings which issue from the apparatus are best observed in the direction of their motion, and should be illuminated from in front or from behind.

In such smoke rings (fig. 44) we may observe the following peculiarities.

(a) Each ring is in the form of a closed vortex tube, its constituent filaments running round in closed curves, that is to say, in circles parallel to the circular axis of the ring, about which the separate smoke particles are seen to revolve.

FIG. 44

(b) The vortex ring moves forward in the same direction in which the air passes through its aperture. This air is being continually drawn from the surrounding atmosphere, and is not charged with smoke. The smoke particles make it clear that it is the same portion of air which takes part in the vortex motion, from the commencement onward, and that none of this original air leaves the vortex, at least during the time for which the motion remains appreciable.

(c) When we consider the smoke ring as a whole, the positive direction is to be determined in accordance with the convention of § 121, that an eye looking in the positive direction along the axis of rotation is to see the particles revolving round the axis in the direction of clock-hands. Since in each cross-section of the ring the motion at the inner part of the circumference is forward, that is, in the direction in which the whole ring progresses, and at the outer part of the circumference is in the contrary direction, that direction round the ring is accordingly to be taken as positive which is seen as a clockwise revolution by an observer looking at the ring in the direction in which it progresses. Compare the arrows in fig. 44.

(d) The form of a ring, especially after it has travelled some distance, is frequently disturbed by draughts of air,

so that one portion of the ring becomes elongated; in this case a simultaneous rapid increase of rotational velocity is observed, though this is soon reduced again by the influence of friction. The phenomenon is an illustration of that constancy of strength' which would characterise a vortex in a frictionless fluid.

(e) If a smoke ring is allowed to impinge perpendicularly upon a wall, such as a window-pane, it does not recoil, but spreads out into a larger circle. This causes an increase in the length of the vortex, and a corresponding decrease in its cross-section, the rotational velocity at the same time becoming greater-noticeably so at the first moment. (Constancy of strength of a vortex J=qw.)

D.-Cyclic motions

In all modern field theories an important part is played by cyclical' motions, gyrostatic and vortical. These motions have been systematically investigated by Helmholtz, with especial reference to the problems of thermodynamics. Hertz has further developed the same conceptions, in his 'Prinzipien der Mechanik,' and has shown how they may be applied to the subject before us.

Just as it is found convenient, in treating of the laws of acoustical phenomena, to commence with some account of the type of motion fundamental to the subject, namely wave-motion; so here we shall find it well to explain in some measure the 'cyclic motions' which we shall have to consider. For the present we shall confine our attention to simple and familiar examples, which may serve to illustrate the nature of these motions and their principal properties. In the second part of the book we shall enter at greater length into their theory, as elaborated by Helmholtz, and further developed by H. Hertz and others.

136. Conception of a cyclic motion. When a system of bodies is in motion, there is in general a progressive change in the distribution of the bodies in space, accompanied by a

change in the condition of the system. Besides this kind of progressive motion, however, there are cyclical or continuously recurrent motions, wherein each particle, on being displaced from its position, is immediately replaced by a similar particle moving in precisely the same manner, so that in spite of the motion the condition of the system does not change.

The uniform rotation of a homogeneous body about an axis of symmetry, and the flow of fluid in a canal of ring-shaped or other re-entrant figure, are examples of such motions (HELMHOLTZ).

Following HELMHOLTZ We shall apply the name 'cyclic motions' to motions of this kind, adopting the following as our definition of a cyclic motion, that is to say, of a 'purely cyclic motion' (compare § 142, below):

Definition. A purely cyclic motion of a material system is such that at each place in the system, when one particle moves from its position, it is replaced after an indefinitely short time by another particle having identical properties, and moving in the same direction with the same velocity.

This definition, taken in its strictest sense, would imply that the motion takes place in an absolutely continuous medium. If this is not the case, if we are dealing, for example, with ordinary ponderable matter, built up of separate molecules, and moving in the manner specified in the definition, we can only regard the motion as purely cyclic in so far as we can ignore the heterogeneity of the matter.

Model of a simple cycle.-A wooden disc, about 10 cm. in radius and 2 cm. thick, is free to turn about a horizontal axis, which is supported on a vertical stand. At unit distance from the axis, a peg is inserted into the disc, to serve as a point of application for the force which is to set the disc in motion. Fastened to the stand, in front of the disc, and pointing towards the axis, is a second peg, which nearly, but not quite, touches the first as the disc goes round, and which serves to mark out a determinate initial direction. Then the position of the disc at any instant is given by the angular displacement of the first peg from the initial direction. If the disc is made to turn with constant velocity about its axis, each position, as it is vacated by

one particle, will be immediately occupied by another, having the same properties and the same motion. If we leave out of account the minute heterogeneity of the material, we may say that the configuration of the whole system remains unchanged during the motion. If the rotation is made sufficiently rapid, it ceases to be evident to our observation, and we have in a certain sense an example of 'hidden motion,' of whose existence we may satisfy ourselves through the medium of the sense of touch.

137. Cyclic co-ordinates and cyclic velocities.-The positions occupied by the constituent parts of any movable system may be defined by fixing the magnitudes of certain quantities which vary with the time, and are called coordinates.'

This term, then, has here a more general meaning than in Cartesian co-ordinate geometry, where the rectangular co-ordinates of a point, for example, are its distances from three co-ordinate planes. A co-ordinate is any quantity (variable with the time) which helps to define the position of a body or of one of its parts, or in any way to determine the configuration of the system, whether the quantity by itself is sufficient to fix the configuration completely, or whether other co-ordinates must be known in addition.

In order to follow as closely as possible the notation. used by Helmholtz and by Hertz, we shall use the symbol p to denote co-ordinates of a system in cyclic motion, so that p then represents a 'cyclic co-ordinate.'

In the example of the model in § 136 the cyclic co-ordinate p will be most conveniently chosen as the ever-increasing angular displacement of the peg in the rotating disc from its original position.

If dy is the change occurring in the value of p during the infinitesimal time dt, then

is the rate of change of the cyclic co-ordinate, that is the cyclic velocity.

In our example, q is the angular velocity with which the disc

rotates.