From these tests it is seen that, in the case of lamps Nos. XI. and XIV., there was practically no change either in the light or in the power absorbed during the first 80 hours of the running at the constant pressure of 100 volts ; and that in the case of the remainder of this batch of lamps the increase in the light and in the power absorbed was comparatively small, the greatest change being with lamp No. XI., where the light rose by about 11 per cent. and the power by about 4 per cent. This is a very different result from that obtained with the Edison-Swan lamps previously tested, and run at a constant pressure of 100 volts, for there the total light given out by the three lamps together rose by 35 per cent. in the first 100 hours of running, and the power by 7:55 per cent. Since, then, as already stated, the vacuum, when tested periodically with the induction-coil , appeared to improve steadily with each of the six lamps Nos. IX. to XIV. during the run of 80 hours, it would appear that improvement in the vacuum was not the sole cause, as stated by Mr. Howell, for the great rise in the candle-power such as we observed with all the Edison-Swan lamps which we tested in 1893 and 1894. It is possible that this rise in candle-power may have been due to a change in the surface of the filament causing the emissivity for heat to decrease, since that would raise the light emitted, as well as the number of candles per watt. Whether such a change in the heat-emissivity of the filament occurs with time we have not yet found out. Another point of difference between this batch of six lamps Nos. IX, to XIV. and those which we tested during the main part of our investigation, is the relative inefficiency of these six lamps. For in no single test at any time during the 80 hours run with these six lamps were the watts per candle less than 4.25, and in many cases they were over 5; while with lamp No. XIII. the watts per candle, as seen in the last table, were as high as 5.15 after this lamp had been run for 13 hours 50 minutes. It is possible, then, that when in the body of the paper we spoke of the group of three lamps which we ran at a constant pressure of 101 volts as being quite abnormal,” because the average watts per candle required with this group during the run of 1340 hours were as high as 4:68, we ought rather to have spoken of the various other groups of lamps which we ran at the constant pressures of 100, 102, and 104 volts respectively as being abnormally good Edison-Swan lamps. At any rate it is clear that it is at present impossible to state the most economical potential difference to employ with any Edison-Swan lamps marked "100 E.F. 8;" since batches of lamps so marked, and therefore nominally the same, really require very different potential differences to be employed to obtain maximum economy with a given price of lamp and of a kilowatt-hour. It may, however, be concluded, from the curves shown in figs. 12 and 13, that the most economical potential difference to employ with any Edison-Swan lamps marked “100 E.F.8," such as have been obtainable in the open market since 1892, is the potential difference that will cause the lamp during a large portion of its life to have an efficiency of about 0.25 candle per watt, corresponding with 4 watts per candle. The value of this potential difference for any batch of these lamps may be determined approximately in the following way. From our tests, already referred to, of eleven EdisonSwan lamps, marked “100 E.F. 8," which had been used in a house for some 200 or 300 hours, we found that the relation between candle-power and pressure was given by Candle-power = ax (volts) 5'91, where a is a constant. Also we found that the relation between candle-power and watts for these lamps was given by Candle-power = bx (watts) ), where b is a constant. Therefore Volts = c *(watts per candle) -258, wbere c is a constant. Hence if one has a batch of Edison-Swan lamps marked “100 E.F. 8,” whose efficiency is, say, 0·222 candle per watt, corresponding with 4:5 watts per candle, when a pressure of 100 volts is applied to them, it would be probably most economical to run them at about 103 volts, since this is the pressure which the last equation tells us will be necessary to raise the efficiency of such lamps to 0.25 candle per watt. From what precedes, it follows that, since the average efficiency of the 100 E.F. 8 Edison-Swan appears to be less than 0-25 candle per watt, the opinion expressed by one of us in an article entitled “ New Lamps for Old,” published in the · Electrician' for September 29, 1893, was correct, viz. that it would be economical to overrun Edison-Swan lamps by applying a pressure about 3 per cent. higher than the marked pressure. And that the reason why this conclusion was apparently negatived by the results of the tests which we carried out during the winter of 1893 and a large portion of 1894, was because the various groups of lamps which we ran Phil. Mag. S. 5. Vol. 39. No. 2-40. May 1895. 2 F at the constant pressures of 100, 102, and 104 volts respectively consisted of specially good specimens. In applying the rule that the economical potential difference is about the one which causes the lamp to produce 0:25 candle per watt, it is important, however, to examine 8-candle 100-volt Edison-Swan lamps when bought to see whether they are really marked " 100 E.F. 8.” For while the result of various purchases of 8-candle 100-volt Edison-Swan lamps during the past three years has always resulted in lamps marked - 100 E.F. 8" being sent us, although the marking on the lamps was never specified by us, a recent batch of lamps that we have received contained among them certain lamps marked “100 B. 8,” which not only differed in the marking but also in the filament being of a simple horse-shoe shape, and not with a loop at the top as in the case of the other lamps. And, on testing these Edison-Swan B lamps, we were surprised to find that with no one of them, when run at 100 volts, did the watts per candle exceed 3.9, and in some cases the watts per candle were as low as 3.01. We have not, however, had these B lamps for a sufficiently long time in our possession to be able to express any opinion about their life-history. XLI. On the Change of Form of Long Waves advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves. By Dr. D. J. KORTEWEG, Professor of Mathematics in the University of Amsterdam, and Dr. G. DE VRIES INTRODUCTION. IN N such excellent treatises on hydrodynamics as those of neglected long waves in a rectangular canal must necessarily change their form as they advance, becoming steeper in front and less steep behind f. Yet since the investigations of de Boussinesq 7, Lord Rayleigh S, and St. Venant || on the solitary wave, there has been some cause to doubt the truth of this assertion. Indeed, if the reasons adduced were really decisive, it is difficult to see why the solitary wave should make an exception*; but even Lord Rayleigh and McCowan t, who have successfully and thoroughly treated the theory of this wave, do not directly contradict the statement in question. They are, as it seems to us, inclined to the opinion that the solitary wave is only stationary to a certain approximation. * Communicated by the Authors. † It seems that this opinion was expressed for the first time by Airy, “ Tides and Waves,” Encyc. Metrop. 1845. Comptes Rendus, 1871, vol. ii. Š Phil. Mag. 1876, 5th series, vol. i. p. 257. || Comptes Rendus, 1885, vol. ci. It is the desire to settle this question definitively which has led us into the somewhat tedious calculations which are to be found at the end of our paper. We believe, indeed, that from them the conclusion may be drawn, that in a frictionless liquid there may exist absolutely stationary waves and that the form of their surface and the motion of the liquid below it may be expressed by means of rapidly convergent series. But, in order that these lengthy calculations might not obscure other results, which were obtained in a less elaborate way, we have postponed them to the last part of our paper. First, then, we investigate the deformation of a system of waves of arbitrary shape but moving in one direction only, i. e. we consider one of the two systems of waves, starting in opposite directions in consequence of any disturbance, after their complete separation from each other. By adding to the motion of the Auid a uniform motion with velocity equal and opposite to the velocity of propagation of the waves, we may reduce the surface of such a system to approximate, but not perfect, rest. If, then, 1+n (n being a small quantity) represent the elevation of the surface above the bottom at a horizontal distance x from the origin of coordinates, we have succeeded in deducing the equation dan an 3 1m2 Dix" at 2 де where a is a small but arbitrary constant, which is in close connexion with the exact velocity of the uniform motion given Ті to the liquid, and where o= = }13 depends upon the depth PI l of the liquid, upon the capillary tension T at its surface, and upon its density p. an On assuming =( we of course obtain the differential at * Though the theory of the solitary wave is duly discussed in the treatise of Basset, the inconsistency of his result with the doctrine of the necessary change of form of long waves seems not to have sufficiently attracted the attention of the author. † Phil. Mag. 1891, 5th series, vol. xxxii. 2 F 2 1 n=kenovia (mod. M= mod. M=VA). equation for stationary waves, and it is easily shown that the well-known equation h n=h sech xa 40 of the solitary wave is included as a particular case in the general solution of this equation. But, in referring to this kind of wave, we have to notice the result that, taking capillarity into account, a negative wave will become the stationary one, when the depth of the liquid is small enough. On proceeding then to the general solution, a new type of long stationary wave is detected, the shape of the surface being determined by the equation h+ h htkl We propose to attach to this type of wave the name of cnoidal waves (in analogy with sinusoidal waves). For k=0 they become identical with the solitary wave. For large values of k they bear more and more resemblance to sinusoidal waves, though their general aspect differs in this respect, that their elevations are narrower than their hollows ; at least when the liquid is not too shallow, in which latter case this peculiar feature is reversed by the influence of capillarity. For very large values of k these cnoidal waves coincide with the train of oscillatory waves of unchanging shape discovered by Stokes*, which therefore in the theory of long waves † constitutes a particular case of the cnoidal form. Indeed the equation I obtained by Stokes, when written in our notation, becomes 277. 3h222 47x n=h cos λ 647273 λ but, as Sir G. Stokes remarks, in order that the method of 22h approximation adopted by him may be legitimate, must 13 be a small fraction. Now, when capillarity is neglected, the wave-length 1 of our cnoidal waves is equal to COS ; a 4K N73 V3(h+k)' * Transactions of the Cambridge Phil. Soc. vol. viii. (1847), reprinted in Stokes, Math. and Phys. Papers, vol. i. p. 197. † Stokes' solution is more general in so far as it applies also to those cases wherein the depth of the liquid is moderate or large in respect to the wave-length. | Stokes, Math. and Phys. Papers, vol. i. p. 210. |