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across the axis of z. This condition may be written, by (1)

h=

= √ ̊ (1—ƒk2 sec2 m2) √ 1 — k32 sec2 ¿mz . dz,

which gives, remembering that k=cos mc,

(28)

mh=π{1-cos mc-fsin mc. sin mc}. (29)

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We may now proceed to evaluate the constants in (1) in terms of h the mean depth of the liquid in the channel. Eliminating U2 between (22) and (13) we get

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If now we solve equations (29), (30), and (31) for m, c, and f, we shall find mh=10025 approximately: hence, remembering that at best the surface-pressure is only to be approximately constant, it will be sufficient to take for our present purpose

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and it is just possible that this may be the exact value. Substituting this value in (30) we get

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(32)

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which differs by less than the experimental error from the value 75 h which I have already {S. § 10} given as a fair average of some experiments I made in connexion with my former paper.

Again, from (13) and (32), or (22) and (34), we obtain for the velocity

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which shows that the maximum wave travels about 25 per cent. faster than low waves in the same depth of channel. Finally, substituting from (32) and (33) in (31) we get

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We have now only to substitute the values of the constants just determined in our general equations.

5. The Final Equations of the Motion.

The fundamental equation (1) may now be written

u+ww=−1·25 √gh{1−·11 sec2≥ (z+x)/h} √√/1—·40 sec2 1(z+ix)/h, . (37) which completely determines the motion of the fluid.

It is convenient, however, to consider the formulæ specially suitable to the regions near to, and fairly distant from, the crest. Thus for regions fairly distant from the crest (where exp(−x/h) is small) (5) and (6) give

and

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− (↓ +18)/Uh= (z+ix) /h+ı 1·24e−(x−ız)/h—&c., . (39)

in which U has the value given by (35).

The equation to the free surface, as given by (7), becomes

n/h=1·04e-x/h — ·44e−2x/h &c., .

and for the pressure-error given by (14) we have

Sp-89 e-2/gph.

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Again, for the neighbourhood of the crest (16) and (17)

give

and

− (u+ww)/U÷·80 √√(5—ix)/h{1+·084(Y—ix)/h} . (42)

(†+18)/Uh=1+•53{ (¿—ix)/h}3/2{1+•50(¿—ıx)h}.

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(43)

We have already seen that the crest is formed by two
surfaces, equally inclined to the bottom, meeting at an angle
of 120°, so that the summit of the wave has the form of a
blunt wedge. We have also seen that in the free wave these
surfaces must be plane or have an infinite radius of curvature
at the crest; we have, however, made no effort to satisfy this
condition, but on substituting the values of the constants in
(20) we find that it gives the radius of curvature at the crest
about equal to thirty times the depth of the water, a result
sufficiently indicating the closeness of our approximation.
The pressure-error near the crest, as given by (27), is

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By (41) and (44) we see that the deviation from constant surface-pressure is everywhere very small; there is a very slight excess near the crest but vanishing at the crest, and a slight defect near the mean level. The deviation has in fact only an appreciable value over a very limited region, say from x=5h to x=1·5h; (41) and (44) are hardly applicable

within this region, but I estimate that within it the maximum defect of pressure is less than that of a head of water, or whatever other liquid the channel may contain, of one tenth of the mean depth.

The accompanying figure shows the form of the wave, only half of it being drawn, however, as the wave is symmetrical

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about the crest. The thicker straight line indicates the bottom of the channel, while the mean depth is shown by a finer line to which the surface approaches asymptotically.

XL. On the Velocity of the Cathode-Rays. By J. J. THOMSON, M.A., F.R.S., Cavendish Professor of Experimental Physics, Cambridge*.

THE neighbourhood of the cathode has been HE phosphorescence shown by the glass of a dischargeascribed by Crookes to the impact against the sides of the tube of charged molecules driven off from the negative electrode. The remarkably interesting experiments of Hertz and Lenard show that thin films of metal when interposed between the cathode and the walls of the discharge-tube do not entirely stop the phosphorescence. This has led some physicists to doubt whether Crookes's explanation is the true one, and to support the view that the phosphorescence is due to ætherial waves of very small wave-length, these waves being so strongly absorbed by all substances that it is only when the film of the substance is extremely thin that any perceptible phosphorescence occurs behind it. Thus on this view the phosphorescence is due to the action of a kind of

* Communicated by the Author.

ultra-violet light, which possesses in an exaggerated degree the property possessed by the ultra-violet rays of the sun of producing phosphorescence when incident upon such substances as German or uranium glass. It is perhaps worth while to observe, in passing, that the light produced in an ordinary discharge-tube by an intense discharge is very rich in phosphorogenic rays. I have been able to detect phosphorescence in pieces of ordinary German-glass tubing held at a distance of some feet from the discharge-tube, though in this case the light had to pass through the glass walls of the vacuum-tube and a considerable thickness of air before falling on the phosphorescent body.

The view, to which Lenard has been led by his experiments, that the cathode-rays are ætherial waves demands the most careful consideration and attention; for if it is admitted, it follows that the æther must have a structure either in time or space. For these cathode-rays are deflected by a magnet, which, so far as our knowledge extends, does not produce any effect on ultra-violet light unless this is passing through a refracting substance: thus if the cathode-rays are supposed to be ultra-violet light of excessively small wave-length, it follows that in the æther in a magnetic field there must either be some length with which the wave-length of the cathoderays is comparable, or else some time comparable with the period of vibration of these rays.

It might be objected that it is possible that the action of a magnet on the cathode-rays is a secondary effect, and that the primary action of the magnet is to affect the main current of the discharge passing between the positive and negative electrodes, and thus to alter the distribution of the discharge entering the cathode: this would affect the distribution of the places of greatest intensity over the cathode, and thus indirectly the distribution of the waves emerging from it. To test this point I shielded the cathode from magnetic forces by means of a magnetic screen consisting of a ring made of soft iron wire the length was about 1.5 inch, its thickness was about 75 inch. When this ring encircled the cathode a magnet was brought up to the tube: the phosphorescent patches inside the ring were not now affected by the magnet, but those on the parts of the tube farther away from the cathode and outside the iron ring were very much displaced by the magnet; thus proving that the magnet acts on the cathode-rays through the whole of their course, and does not merely affect the place on the cathode at which they have their origin. There thus seems no escape from the conclusion that the establishment of the hypothesis that the cathode-rays

are ætherial rays would also prove the finiteness of the structure of the æther.

The following experiments were made with the view of determining the velocity with which the cathode-rays travel, as it seemed that a knowledge of this velocity would enable us to discriminate between the two views held as to the nature of the cathode-rays. If we take the view that the cathoderays are ætherial waves, we should expect them to travel with a velocity comparable with that of light; while if the rays consist of molecular streams, the velocity of these rays will be the velocity of the molecules, which we should expect to be very much smaller than that of light.

The method I employed is as follows:-The discharge-tube was sealed on to the pump, and the two electrodes were placed at the neck of this tube. The discharge-tube was covered with lampblack, with the exception of two thin strips in the same straight line from which the lampblack was scratched: these strips were about 10 centim. apart; the one nearest to the negative electrode was about 15 centim. from the electrode, the other was 25 centim. from the electrode. They were chosen so as to phosphoresce with, as nearly as could be judged, equal brilliancy when the discharge passed through the tube.

The light from the phosphorescent strip fell upon a rotating mirror about 75 centim. from the tube. This mirror is the one used by me in my experiments on "The Velocity of Propagation of the Electric Discharge through Gases" (Proc. Roy. Soc. 1890), and is described in that paper. The only change made in the mirror was to replace the single plane strip of silvered glass which was used in the previous experiments by six strips of mirror fastened symmetrically round the axis. The mirror was driven by a large grammemachine.

The images formed by reflexion from the mirror were observed through a telescope, of which the object-glass was a large portrait photographic lens of 4-inch aperture, the eyepiece a short-focus lens: when the mirror was at rest the two images of the phosphorescent strips were seen in the same straight line, and the adjacent ends of the two images were brought into coincidence by inserting between one of the phosphorescent strips and the mirror a very acute-angled prism. The point of the experiment was to see if the images of the two phosphorescent strips remained in the same straight line when the mirror was in rapid rotation. If, for example, the cathode-rays travelled with the velocity of sound, they would take about of a second to pass from 3300 one strip to the next; if the mirror were rotating 300 times

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