May io, 1894J 



NA TURE 



45 



houses. The optical problem of the lighthouse engineer is to 

 construct apparatus which shall usefully direct all the light 

 produced. The present forms of apparatus are in their leading 



' features due to Fresnel, the able mathematician, who established 



' on an absoluiely firm foundation the undulatory theory of light. 

 To properly design an optical apparatus formulce must be used, 

 and the advantage is great if the designer can with ease manu- 



I facture the formulae he requires. 



Submarine telegraphy yields some interesting examples of the 

 application of the higher mathematics. When a cable across the 

 AtKintic was first seriously entertained, the first paint to be 

 settled was, how many words a minute could be sent through 

 such a cable. This w.as the most practical que^tion possible. 

 Upon the answer depended the prospect of the cable paying 

 commercially if successfully laid. The matter was dealt with 

 by Prof. Thomson,' of Glasgow, now Lord Kelvin. He showed 

 that the propagation of an electric disturbance in a cable 



I could be expressed by a partial differential equation, and that 

 the solution of this equation under certain conditions applicable 

 to practice could be expressed either by a definite integral or 

 by an infinite series. The values of these were calculated, 

 and hence before an Atlantic cable was laid at all it was 

 known how long it would take a signal to reach the opposite 

 shore, and how much its intensity would be diminished in 

 transmission. Referring to Fig. 5, abscissae represent time, 

 reckoned from the time of making contact at the sending 



Fig. s. 



;nd of the cable, ordinates the currents at the receiving ends, 

 ;urve (l) gives these currents when the contact at the receiving 

 nd, after being made, is continuously maintained. It will 

 5e observed that for a time a there is hardly any current at the 

 eceiving end, that then the current rapidly increases and attains 



half its final value after a time equal to about 5a. Curves 

 l) .... (7) show the currents at the 'receiving ends when the 

 ;ontact is made at the sending end maintained for times a, za 



. . 7a respectively, and then broken. Looking at curve (1) 

 me sees how small is the amount of current and how long it 

 asts compared with the time during which cont.act is made. 

 The time a depends on the length and character of the cable ; 



t is equal to k c fl loge ^/ir-, where i is the resistance per unit 



ength, c the capacity per unit length, and / the length of the 

 :able. The knowled/e of what is the commercial value of a 

 able depends on a knowledge of the value of a, and this cannot 



■e obtained without knowing the differential equation c i — 



j,- to which I have referred, and its by no means simple 



olution either as a definite integral or as an infinite .series. So 

 ar as I know, this piece of higher mathematics cannot be 

 vaded by any mere elementary treatment. The transmission 



' " M.ilhem.vical .ind Physical P.ipers," vol ii. p. 61. Sir \V. Tfiomson. 

 '^ V is ihcpotenlial, t the time, and x the distaQCe from the sending end 



1 the cible. 



NO. I 280, VOL. 50] 



of disturbance in a cable is quite different from the transmission 

 of sound waves in air, which move with constant velocity. If 

 the cable be doubled in length, it takes four times as long for the 

 signal to pass through it instead of just twice as long, as would 

 be the case if it were a proper wave motion. In fact the time 

 of passage between the making of contact at the sending end of 

 the cable and the beginning of the resulting disturbrnce at the 

 receiving end, varies as the square of the length of the cable. 

 The mathematical theory is exactly the same as that of the 

 transmission of heat in a iilate, one surface of which is suddenly 

 exposed to a temperature different to the temperature of the 

 plate. This is constantly occurring in the application of 

 mathematics — one piece of mathematical work serves for many 

 physical problems having apparently little in common. Fourier 

 long ago discussed the heat problem, little dreaming that his 

 analysis would be just what was wanted for ascertaining how 

 fast signals could be sent across the Atlantic by a system of 

 telegraphy which in his days had not even been projected in its 

 simplest form. The ;same differential equation also gives the 

 theory of the transmission of telephonic messages through 

 cables ; but the solution is then easier, and tells us exactly why 

 it is so much more difficult to speak through 100 miles of cable 

 than through 1000 miles of overhead line. As I have just 

 stated, the differential equation of the disturbance in the cable 



>s c ^ -jr= -7-^. A musical note of period T spoken into the 



cable through a telephone is properly represented 



by A sin " — ; the disturbance in the cable will 



be— 



as may be easily verified by differentiating. This 

 equation tells us everything. It tells us the rate 

 at which the waves diminish with the distance. 

 This rate increases with the resistance, with the 

 capacity and with the frequency. If the capacity 

 is at all considerable the diminution is rapid. 

 The velocity of the waves is not the same for all 

 frequencies, as is the case with w.aves in air, but 

 varies as the square root of the period, so that if 

 two notes were sounded the high note would 

 arrive after the low notes, and the resultant effect 

 would be entirely destroyed. Here, again, it is 

 difficult to see how the difi'erential equation and 

 its solution can be evaded. 



Though the history of the telegraph dates only 

 from a little more than fifty years ago, it is ancient 

 in comparison with the other great applications 

 of electrical science, which have received their 

 development during the last fifteen years. Here again 

 mathematics which are not quite elementary have played 

 their part. In the theory of transformers we find another illus- 

 tration of the need of knowing how formulce are obtained if 

 they are to be correctly applied. The early transformers were 

 made with unclosed magnetic circuits ; there was an iron core, 

 but the lines of magnetic force passed through nir lor a consider- 

 able part of their path. In this case a complete mathematical 

 theory was not very difficult. But speedily closed magnetic 

 circuits were found to be better, and the relation of magnetic 

 induction and magnetic force became all important. If anyone 

 were to apply mathematical formula;, which were true for the 

 earlier transformers, to the later ones, his results would be 

 inaccurate. Indeed a wholly different method of attack on the 

 problem was needed, taking account of the facts as they are, 

 and not applying results which were true of older apparatus to 

 cases essentially distinct.' 



The employment of alternating currents has brought into use, 

 as a necessity for understanding the actually observed pheno- 

 mena, a great deal of mathematics. Why is the apparent 

 resi-stance of a conductor greater for an alternating current than 

 for a direct current ? And by resistance I do not mean the quasi- 

 resistance due to self-induction." The mathematical electrical 

 theory is ready with .an answer ; it is ready, too, to tell us how the 

 difterence depends upon the frequency of the current and on the 

 size of the conductor. In the case of a cylindrical conductor 



' Procftiitntrs of Royal Society, February 17, 1S87. 

 - Lord Rayleigh, P/iil. Mag., vol. x.\i. p. 3S1. 



