Electromagnetic Theory, Vol. 2 (1899) — part 28 of 31

But, very curiously, there has been an exception to the rule. There have not been wanting electricians who, when telephony was extended to circuits containing underground work, sup- posed that high, not low resistance, was good for telephony. This was giving up the law of the squares with a vengeance. But there was really no warrant for such a conclusion, either theoretical or practical As this view turned out manifestly wrong by practical failure, it was followed by a precipitate re- vulsion, the law of the squares being again seised, and grasped more firmly than ever, and against all reason. The fact that R

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and S are prejudicial does not make the law of the squares. To have that, K and S must be the only controlling electrical factors.

All the other effects are of later recognition, and were mostly discovered, and their laws mvestigated, and consequences worked out by myself. They completely remodel the subject. The old views remain rights in their proper application, but have a very limited application in general, as in modem tele- phony particularly. My extended theory opened out a wide field for possible improvements in signalling over long circuits in advance of what could be possible were B and S the only controlling factors, and so far as it goes, modern telephony already goes far beyond the supposed restrictions of the law of the squares.

We should not only consider the effects of varying the four line-constants one at a time, but also of combinations. Thus, K may be counteracted by L. The former is beneficial, inas- much as it lessens the distortion, but it is prejudicial in the attenuation it causes in transit. But if we simultaneously in- crease L, we counteract the attenuation caused by K ; more- over, we still further lessen the distortion. This is corrobora- tive of the preceding remark as to the unreserved benefit due to L. We see, therefore, that [K] and [L] above point the way to salvation, if that be by means of increasing the distance through which signalling can be carried on with a cable of given type as regards ii and S. But let it not be forgotten in studying £ and L, that B and S themselves might perhaps (in some particular scheme, for example) be reduced as cheaply as K and L could be carried out. Be liberal with the copper, for one thing. Look out for some practical way of insulating the wires which shall not make the permittance be so great as it is at present with gutta-percha or india-rubber. Something of this sort has been done for telephone underground cables, though I do not know the practical merits. But, as regards Atlantic cables, it should be something very good and trustworthy, or the reduced permittance may be bought too dearly.

There has been some difficulty in getting people to grasp the ideas embodied in [K] and [L], especially the latter, of course, on account of its novelty ; whilst [K] was already partially recognised by knowing ones, and is capable of very ready ex- planation. [L] was the trouble. Now, remembering the state

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of fog that prevailed in the fifties as regards the effects of R and S alone, there is no reason for wonder that more recon- dite effects, as of L, unaccompanied by prior experience, should not be readilj reoognised when pointed out, though that was no exooae for burking the matter. [K] and [L], especially the latter, require to be studied to be understood. But if people will not take the trouble to study the matter, how can they expect to understand it 1 Now I find that there has been too much of an idea that I have merely given some very oompli- oated formula which no fellow can understand, and have made some dogmatic and paradoxical statements about the conse- quences. But a reference to my Electrical Papers ^ will show that I have treated the matter of propagation yeiy fully, in the elementary parts as well as in the advanced, and from many points of view, and that I have not been above pointing the moral by numerical examination of ijBsults, whilst at the same time I have been particularly careful in elaborating the theory of the distortionless circuit, on account of its casting so much light upon the subject generally. The notion that has been promulgated, that I cannot niuke myself understood by physicists, recoils upon its autliors with great force. Letting alone the fact that some physicists have understood me, and moreover, the fact that various students have understood me (as I know by my correspondence), there is the very notable fact that the promulgators of the false idea (which must be most discouraging to students) have evidently not taken the trouble to examine my writings to see what my views really are, but have only made acquaintance with fragments thereof, I have advertised the distortionless circuit so often that I have become ashamed of doing so. Nevertheless, since it is, as I have often pointed out, the royal road to an inner under- standing of the subject in general, I do not hesitate now to add some more remarks in connection therewith, and leading up thereto, in attempted elucidation of former work. We should first consider [K] alone, or in conjunction with [R] and [S], and then [L], and finaUy both [K] and [L].

The Beneficial Effect of Leakage in Submarine Cables,

^212. What leakage does is to increase the attenuation of wiyves in transit^ but at the same time to lessen the distortion

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they mdEat. Now there is, elementary knowledge, no difficulty in seeing why leakage on the line should diminish the strength of the received current, in spite of the fact that the strength of the current sent is increased by the leakage. Ohm's law applied to a simple circuit with a shunt attached will suffice to give the reason so far as one fault is concerned ; and the principle is the same for any number of faults, or for dis- tributed leakage.

As regards the lessening of the distortion, that is not quite so elementaiy. It does not present itself for consideration at all in common telegraph circuits. When, however, a cable is in question, and there is manifest distortion of signals, the influence thereupon of leakage becomes an obvious matter of inquiry. We may do it on the basis of the electrostatic theory. In fact, in his telegraph theory, Lord Kelvin did insert a term in the differential equation to express the eflfect on the poten- tial of uniform leakage, though it was merely in passing, and I do not think there was any development of the matter in his later papers. The obvious attenuating effect of leakage, and the manifest evils of faults, caused high insulation to be aimed at, and possibly caused the full efl'ect of leakage to be over- looked, and its theory to remain uninvestigated.

Now a cable is a condenser, or leyden, and a very big one too, although its permittance is so widespread that the per- mittance of a yard or so is very small. The discharge of a condenser against resistance takes time, and so does its charge, and the greater the resistance the greater the time needed. In the extreme, a perfectly insulated charged con- denser keeps its charge unchanged, and so does a well-insulated cable, if also insulated at the ends. Even when it is earthed (or grounded, as the Americans say) at both its ends, only the portions close to the end can discharge at once. Away from the ends, the discharge has to take place through resistance, to a varying extent though in different parts, when we regard the long cylindrical condenser as a collection of small con- densers side by side in parallel, with all the similar poles joined together by resisting wires. This is how it comes about that the resistance acts conjointly with the permittance in causing '* retardation " in cables, so that the electrostatic time-constant is proportional to their product, or BSP» It is obvious, then,

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"tbat we can facilitate the discharge by lessening the resist- ance to discharge, that is, by leakage in the cable, either dis- tributed or in detached lumps. At the same time the charging is similarly facilitated or expedited.

Now, in making signals from end to end, the operation of charging the cable is necessarily precedent to that of receiving current at the distant end, which, in fact, represents a part of the charge coming out. Therefore, to expedite changes in the charge in the cable, not necessarily or usually complete charges or discharges, results in expediting changes in the current at the distant end. That is, it lessens the distortion at a given working speed, and so expedites the signalling. There is, in the electrostatic theory, no limit to this process, and by suffi- ^e&tly reducing the insulation resistance, or, which is the same, increasing the leakanoe (which is, perhaps, an admissible short name for leakage conductance), we might signal through the cable as fast as we pleased, were there no other effect produced. This other effect is the extra attenuation. Whilst the possible speed of signalling can be raised apparently ad lib,, there is only an infinitesimal current left to do it with. (Bemember that the influence of L is not now in question.) tt is important to recognise that it is a consequence or natural extension of the <old electrostatic theory, without any appeal to self-induction, that you can signal as fast as you like through an Atlantic cMe of the present type (with leakage added, of course) simply by throwing away most of the current, and utilising the small residuum that manages to sunrive the ordeaL

Thus, a proper Atlantic cable, suitable for rapid signalling, should not have high insulation resistance, but the very lowest possible consistent with getting enough current through to work with, and other considerations, mechanical and com- mercial, &c. The theory, moreover, is a sufficiently clear one, and does not need elaborate mathematics to make it at least probably accurate, though it may, of course, be desirable to thoroughly verify the consequences mathematically.

It is now about 23 years since I came to see, substantially as above, the theoretical absurdity of a system of signalling which deliberately and seemingly of purpose places in the way the greatest possibly hindrance to the efficient trans- mission of sigpmls, viz., by the use of the highest insulation

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poaiUe, oaufling them to be regularly strangled, ehoked, and matilated, to say nothioi^ of their aometimeB dying of inanitioa.

Bat this was mere theory. Am a matter of fact^ the effeofr of leakage presented itself to me at that time in pcaetioe on real liye cables in a surprisiDgly dififorent manner. The opposition between plausible theory and live praotioe was so glaring that it will be instructive to briefly describe it nexti

Short History of Leakage Effects on a OaUe Oiieolt^

§213. First, as regards long land lines, of say 400 miles in length (which is a long distance in this small country), leakage presented itself as an unmitigated nuisance. The worst case of all was during a continuous soaking rain all over the country. When this had gone on for a few hours the insulation resistance fell so low, by leakage over the dirty wet insulators mostly, though assisted by contacts with wet foliage, <fec., that many of the important circuits were completely invalided, the received current (even with increased battery power) being too small for practical use with the instruments ooncemed. This was usually the case in the U.K. on any very wet day. But it was also- the same in effect in the E. and I., although not so many lines were invalided, nor did they break down so quickly.

Incidentally, it may be mentioned here that the so-<salled> "earth-currents'' (meaning currents tn the line itself), occurring during magnetic storms, such currents being a manifestation thereof did not seem to mind the leakage much, perhaps because their E.M.F. was not like that of a battery at one end of a line, but acted continuously along the circuit, by a continuous slow change going on in the earth's magnetic foroe^ or, rather, in the magnetic force in the air surrounding the earth. Probably the "earth-current" goes the same way ia the line as in the earth (provided the line is not set at right angles to the real earth-current), whereas when a battery is> used, the earth and line-currents are opposed. But it is not easy in wet weather to get a good magnetic storm, so that dose observations can be made regarding the influence of leakage on earth-currents. We may dismiss them, and sum up that, as regards land-lines, leakage does not seem to have a single redeeming feature.

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The effects of leakage next presented themselves to me on «able cirouits with land-lines attached, and were exceedingly curious. There were two circuits of the kind, but one in par- ticular had a varied history as regards leakage. It was a cable of about 400 or 450 miles running east and west, having a land line of 120 miles at the east end, and one of 20 miles at the west end. The apparatus used was the Wheatstone automatio, that is, an automatic transmitter and a polarised receiver, reooiding dots and dashes, and the working was done without eondensers, by utilising the earth direct to complete the oiionit.

The best results were obtained in fine dry weather with a hsxd frast-^e harder the better. The insidation of the long (east) land line was then at its highest^ and the signalling could be done up to 30 words per minute or a little more from west to east, though connderably less the other way, for a peculiar reason. But this was quite exceptional, and did not last long. The next best was in any very dry weather. But in ordinary «Terage fair weather, not rainy or damp, the speed was much lower, say 24 or 20 words per minute. The effect of rain was always to lower the speed, perhaps down to 15 words per cmnutQ. A leakage fault on the land line or a contact with another wire had a similar effect, only worse, provided it was a bad fault, and the speed would go down to 10 words per minute or less. Now it is possible that the extra good results obtained in a hard dry frost arose partly from the reduced resistance of the east land line (the usual iron wire), but there was no doubt at all that the main cause of the wide variations of speed was leakage on the land lines, especially on the long one, of course.

Now it is to be noted here that in the former case of long land lines the leakage was prejudicial by the attenuation it caused, whereby the received currents were made too feeble to suit the apparatus. The working speed, though relatively fast, was too slow for any notable distortion of signals in transit by the condenser action so marked in cables. But in the latter case, of a cable with terminal land lines upon which similar leakage occurred, the prejudicial effect was not due to the attenuation. There was plenty of current — in fact, a super- abundance— ^but it did not come out at the receiving end properly shaped for making dots and dashes, except at a greatly lowered speed. It should also be understood that the

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actual speed of working was always pushed up to the greatest possible (the press of business being such as to make 25 hours*' work per day all too short), and that the automatic character of the signalling made the fixation of a limiting speed, without complete failure of any marks, quite definite at any particular time. Practically, however, the working speed was pushed & good deal higher than this limit by making regular use of mutilated signals. Now it was proved by daily fizperienoe that there was a remarkable sensitiveness to damp, or leakage of any kind, the speed of working, conditioned bj the reoeptioa of good marks, going up and down like the barometer, though with a far wider range than that instrument. The east land line was so manifestly the offender that it oame in for a great deal of blame. Yet it was not a bad line by any means, and its leakage would have passed unnoticed, had there not beeik such a peculiar lowering effect on the speed connected there- with.

But how about leakage in the cable itself, seeing that the- above results were so out of harmony with what they should have been, by general reasoning regarding retardation f This- question was soon answered by the automatic development of m first-class leakage fault in the cable. Its effect on the speed was substantially the same as that of leakage on the land-line. It lowered the speed, and since it was always on, the effect was- permanent. The ordinary fluctuations disappeared in a great measure, as of course did the fine weather high speeds. It wa»^ like continuous heavy wet. As the fault got very bad the speed went down to 10 words per minute or less, in spite of increased battery power.

On describing this prejudicial effect of leakage to the elec- trician of the cable ship which came to remove the fault (it was the late Mr. S. E. Phillips, of Henley's), he assured me I was quite wrong. It was impossible for a leak to act like that. So- said the captain, too. It was known, said they, that a leak was an excellent thing. It had been found so on the Frencb Atlantic not long before, and in other cases. I fully believed this, because it seemed theoretically sound, and I had heard similar statements before. But they did not assist me to an understanding of the patent fact that leakage could be highly prejudicial instead of beneficiaL In corroboration of this, wheik

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the fault was removed, the speed went up agaun, and the normal state of things was restored.

The next stage in this history was the arrival of that won- derful instrument, Thomson's recorder, which had been recently brought out. It was the original Atlantic cable pattern, with 20 big tray cells for the electromagnets, and a batch of smaller ones for the monsemill and driving gear. This instrument ■implified matters greatly, for the ourrent oould be considered as and seen to be a oontinnous function of the time, whereas with the polarised recorder of Wheatstone (an instrument quite VDSoitable for cable work) the variations of the ourrent oonld be only roughly guessed.

Now, a peculiarity of the siphon recorder was that no definite limiting speed was fizable. If one person thought 40 words per minute fkst enough for good marks ; another, more prac- tised, would as easily read them at 50 or 60, and even manage to make them out at 70 words per minute, though the last was imdoubtedly rather troublesome. The automatic transmitter being still used, the dashes became big humps, and the dots little ones, on the recorder slip, and at 70 words per minute, although the big humps were plain enough, the little ones were mostly wiped out, though it could be seen where they ought to have been, so that some reading was possible. The absence of a definite limiting speed under given circumstances introduced a fresh complication. Nevertheless, a careful study of the recorder slips taken at various speeds, but particularly in the neighbourhood of the critical speed for the polarised recorder, proved that the plain result of general reasoning that leakage should be beneficial in general was certainly true, and, more- over, furnished a partial explanation of the anomalous behaviour of the polarised recorder.

The siphon (Thomson) recorder, with its shifting zero, was lublimely indifferent to the vagaries of the current which were ■0 destructive of intelligible marks in a dot and dash instrument^ ■o that ordinary leakage passed quite unobserved as a rule. Nevertheless, it was plainly to be seen that when there was much leakage the result was to improve the distinctness of the marks by reducing the range of variation of the current from the aero line, and improving the dots, or small humps, relatively to the dashes, or big humps, which were reduced in sise.

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But, to ollnoh the matter, a aeoond leakage fault appeared in the cable whilst the eiphon recorder was on, and rapidly deyeloped. Now, it was only at the west end that there was a siphon recorder ; at the east end was the polarised recorder. The result was that the speed from west to east went down greatly as the fault developed, in accordance with prior know- ledge. But the other way, from east to west, receiving with the siphon recorder, the working was greatly improved. The ultimate result was that when the fault got very bad no marks at all could be got at the east end, even by key -sending, owing to the great attenuation, whilst we could receive first rate on the siphon recorder at the west end. On the very last day, when communication from west to east had to be done by a roundabout course, we were able to receive a long batch of, 1 think, 150 messages at high speed (about 55 or 60 words per minute) by using 80 Leclanch6 cells at the sending end, and putting on the electromagnets all the tray batteries available, including most of the small ones, to ma^ify the marks. Now this might seem to be merely a testimony to the sensitiveness of the instrument. But the significant point was that although the marks were very small, in spite of magnification by increased battery power at both ends, they were exceedingly clear. They were the best ever got at the speed. We were practically working past a fault of no resistance, for the cable soon gave up the ghost. The tail end of the big batch finally degenerated to a straight line, indicative of a very dead earth.

On the removal of this fault, the received current was largely increased of course. But, simultaneously, the distortion of the marks on the siphon recorder became huge at the speed 60. It needed to be much lowered to give conveniently readably marks.

Explanation of Anomalous Effects. Artificial Leaks.

§214. Now the above facts gave a decisive and cumulative

proof that leakage was beneficial even when concentrated at a single spot, provided the current- variations were recorded in a simple manner. But a complete explanation of the very large decrease in the speed possible when the polarised recorder w is used, turned out to be rather a complicated matter in detail, although sufficiently simple in the fundamental principle.

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This effect, so detrimental to an efficient use of the cable, was mixed up with another one, viz., a difference in the working speed according to direction, of say 25 per cent, (with the same instruments), but variable with ciroumataiices. This effect, again, seemed to have two causes, one depending upon the circuit, and the uncentrical position of the cable in it, indepeik* dent of the kind of instruments used ; whilst the other was special and peculiar to the automatic transmitter and receiver when combined with the uncentrical position of the cable. It may be readily imi^ned that the resultant effect on the working speed requires some study to be understood. It will be sufficient here to mention the instrumental peculiarly in its relation to the detrimental effect of leakage.

The original automatic transmitter of Wheatstone was con- structed to make dots only upon the slip of a special reoeiver ; there were two rows of dots, one set being for the dots, and the other for the dashes of the Morse code. But when the luTentor wanted the E. and I. Co. to take it up, they demanded dots and dashes, as usual. So the instruments had to be modified. Now I have personal reason to know that a few years later the inventor was quite at sea as to the reason of the peculiar effects observed in working his instruments on composite circuits and cables. It is therefore pretty safe to assume that previously, when the conversion was made, the manner of the conversion, from making double dots to dots and dashes, was dictated purely by considerations of mechanical convenience, and this conclusion may be made nearly certain by a comparison of the original with the modified transmitter. But Mr. Stroh knows all about it. At any rate, the peculiar way of making the dash and the long space was the cause of all our trouble with leakage, which cost the Company an immense sum of money, viz., what they might have gained otherwise, for they were blocked with work, and might have had much more.

To make a dot, a short positive current was sent, immediately followed by a similar short negative current. To make a dash, after sending the first short positive current, there was an interval of no current lasting twice as long as the first current, followed by a short negative current which terminated the dash. Similarly, to make a long space, a short negatiye ounent oame firsts then an interval of no current, and finally a short

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positive current. Now in these intervals of no current, the sending end of the line was insulated. When sending current, on the other hand, the line was to earth, through the battery. Two distinct electrical arrangements were therefore operative in turn, viz., a cable with both ends earthed, and a cable with only one end (the receiving end) to earth. If the former sys- tem always prevailed, the manner of transmission of electrical changes would follow one law. If the latter system prevailed, it would follow another law, and would give much greater retardation. In real fact, the two ways were in alternate action irregularly. This has to be considered carefully in examining the influence of the uncentrical position of a oable in a compo- site circuit, and also of leakage.

Now the practical effect worked out thus. There was a real poBitiye and material advantage in making the dash and long space of the Morse code by a short initial current instead of a long one, as in ordinary working with reTeming keys. I esti- mated the gain as amounting to from 100 to 125 per oentw under fibvourable curaumstances. This implies that the limiting speed is really worked up to. But it was iritally necessary to secure this advantage, that the insulation should be retj good. A peculiar balance had to be preserved between tli* positiTC and the negative currents. The currents were of equal duration and of alternate signs, but necessarily not equidistant in time. Now, with exceedingly good insulation, the short oorrent sent at the beginnuig of a dash was sufficiently prolonged in reception as to not merely make a good dash firmly, but to keep the balance for the next currents. But if you put a fault on the line, and let the charge leak out during the dead interval of the dash, then the next following negative current, terminating the dash, would be too strong, so that an imme- diately succeeding positive current (with a negative current after it) to make a dot would miss fire altogether. Similar remarks apply to the space. If the negative current that initiated the space were allowed to die away by the action of a leak, the next following positive current would be too strong, and cause an immediately following negative current to miss fire. Remember that the reception of the signals was with a polarised recorder, the zero of which could not be continuously altered to catch the missmg currents except by a very artful person who knew what

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was coming, and was quick enough to alter the regulation exactly at the right moments. It will, by the above, become roughly intelligible how a leak, actually quickening signals in a zemazk- able degree, can yet be most prejudicial in a special system of signalling requiring the presei^tion of a sort of balance. If the siphon recorder be substituted the necessi^ of the balance disi^pean ; but the marks would be yery hard to read at the same speed. To make them readable the speed should be at least doubled. The dashes will then become well-formed humps.

The reader who may desire to pursue the subject of leakage effects further may be referred to my "Electrical Papers,** Vol. I., especially p. 53 for continuous leakage, p. 61 for more details relating to the aboTC-described case, and p. 71 for the

general theory of leaks, which is fully worked out in seyeral

cases to show the expediting effect. I also there propose arti- ficial leaks. The following extract (p. 77) is easy to read, and so may be worth quoting :—

** When a xuttoral foult, or local defect in the insulation, is developed In a cable, it tends to get worse — a phenomenon, it may be observed, not oonfined to cable^faults. Under the action of the current the fault ia

increased in size and reduced in resistance, and, if not removed in time, ends by stopping the communication entirely. Hence the directors and officials of submarine cable companies do not look upon faults with favour, and a eharp look oat is kept by th» faolt-flndem for their detection and mbeequent removaL Bat an artificial ikalt, or connection bj means of a coil of fine wire between the conductor and sheathing, would not haye the objectionable features of a natural fault. If properly constructed it would be of constant resistance, or only varying with the temperature, would contain no electromotive force of polarisation, would not deteriorate, and would considerably accelerate the speed of working. The best position for a single fault would be the centre of the line^ and perhaps ^ of the line's resistance would not be too low for thefMilt^"

The effect of one leak is marked, perhaps 40 per cent. What does this represent in £ sterling ? How many thousands or tens of thousands of pounds' worth of oopper and g.p., dco., would be equivalent f If one leak would not quite do it, then two would.

But to have only one or two leaks is to look upon the leak as a dangerous nuisance. Although the beet position for a given amount of leakance in a lump is in the middle of the

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cable, yet to have the given leakauce in a lump is the worst arrangement as regards its accelerative efficiency. The best arrangement is uniformly-spread leakance, as much as possible, with, however, a thoroughly useful residual terminal current to work witli, which current could be varied far more rapidly than is possible with a single leak.

Is it impossible to find an insulator of comparatively low resistance which should be suitable in other respects 1 Tele- graph cable manufacturers would probably say, No. That, however, need not be considered concluuve^ because, as is well known, when an industry or institution is onoe established, it always gets into grooves, and has to be moved out of them hj external agency, if at all. But should such an insulator be un- attainable, then the only alternative is to have artificial leaks, as many as possible. The theory is, I think, quite plain, and I know that the practice is also^ up to a certain point. But here be grooves again. For people working in established ways have their own proper work to do, and have no time to waste upon '* fads." So much is this the case that it often happens that they have exceedingly little knowledge of how things would work out if they departed from the ruts they are accustomed to run along. That is to say, the spirit of scientifio research, which was to some extent present in the industry in Its early stages, has nearly all evaporated, leaving behind regnlar rules and a hatred of fads. I am no wild enthusiast, having been a practician myself, but it has certainly been a matter of somewhat mild surprise to me that cable electricians, who have had such unexampled opportunities in the last 20 or 30 years, say, should have done nothing in the matter of leaks. Perhaps there may turn out to be objections at present un- thought of. The same could be said of anything untried. Several cases of resistance to change of established practice, where the changes ultimately turned out to be of great benefit, have come under my personal observation. The proposed changes were (I believe quite honestly) first scouted as lads, and the innovator was snubbed ; next, under pressure, they were reluctantly tried ; thirdly, adopted as a matter of course, and the innovator snubbed again. But this is the nature of things, and can be seen everywhere around — ^very notably in the political world.

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gelf-Indaction imparts Momentma to Waves, and that carries tliem on. Analogy with a Flexible Oord.

§215. We have now to consider in what way self-induc- tion comes in to influence the transmission of waves along wires. The first step in this direction was made by Kirchhoff as long ago as 1858, but it is only in recent years that self- induction has become prominent as an active agent, and its real resultant effect worked out and understood, more or leas. This effect is, on first acquaintance, a somewhat sur- prising and paradoxical one. Everyone knows the impeding effect of self-induction in apparatus. How, then, in the name of Faraday, can it be beneficial in a telegraph or telephone oirouit, and make signals bigger and clearer ? To answer thia^ it will be oonyenient to make use of an analogy in the first place.

There is, in a great measure, a formal resemblance between the problem of a telephone circuit alraig which electromagnetio disturbances are being propagated, and the mechanical problem of the transverse motions of a stretched flexible cord. Bat to make the formal resemblance be also a practical resemblance, aereral little things have to be attended to, and reserrations made. Now there are a great many mechanical problems con- cerning wave-motion subject to friction which make better analogies for the use of the mathematical inquirer ; but the case of a flexible cord is more or less familiar to everyone, and is, up to a certain point, easily realisable.