i6 



NA TURE 



[May io, i'^94 



the sotalion involves a knowWge of Bessel's fanclions. We 

 learn that if the current has a high frequency, or if the con- 

 ductor be large, there will he very little current in the centre of 

 the cylinder, and that therefore for any practical purpose the 

 centre of the cylinder m'ght just as well not be there ; the cur- 

 rent is largely confined to the part of the conductor near to its 

 surface. The currents at different depihs in the conductor 

 attain to their maximum values at different times ; those near 

 the surface of the cylinder occur before those at jome distance 

 from the surface. The mathematical conditions are expressed 

 by the same equation as is used to express the disposition of 

 heat in a cylinder the surface of which is submit'ed to a periodic 

 vaiiaiion of temperature. Anyone who had thoroughly mastered 

 the heat problem would be quite prepared to deal with the pro- 

 blem of currents in a conductor. It cannot be too often 

 repeated, any piece of pure mathematics which finds one appli- 

 cation to a physical problem is almost sure to find, in exactly 

 the same form, applications to other problems which super- 

 ficially are absolutely distinct. The differential equation in 



this case is ^f - =( — _. j ), the similarity of physical 



dl \di^ rdr ] ' *^ ' 



condition to the problem of linear propagation of heat is close, 



bat the mathematics differ materially owing to the presence of 



I dv 

 the term -, y- in the equation. Mathematics deals with the 



relation r.f quantities to each other without troubling as to what 

 the physical meaning of the quantities may be. Hence it is 

 that the mathematical treatment of two such problems as the 

 distribuli"n of currents in a cylindrical conductor and of heat in 

 a cylinder is identical, whereas the treatment of the distribution 

 of heat in a cylinder is quite distinct from the treatment of the 

 distribution of heat in a sphere or in a solid bounded by two 

 parallel planes. 



A curious phenomenon was observed in the large alternate- 

 current m.achinesat Depiford when connected to the long cables 

 intended to take the current to London. The pressure at the 

 iDachines when connected to the conHuctor-j was, untler certain 

 conditions, actually greater than when not so connected. The 

 phenomenon is one of resonance very analogous to the heavy 

 rolling of ships when the natural period of roll is about ihe 

 same as the period of the waves.' The period of the alternating 

 current corresponds to the period of the waves, the self-induc- 

 tion of the machine to the moment of inertia of the ship, the 

 reciprocal of the capacity to the stiffness of the >hip, and the 

 electrical resistance of the conductors to the frictional resistance 

 to rolling. The mathematics in the two cases is then the same. 

 The effect was predicttrd long before it was observed in a form 

 calculated to cause trouble. 



A problem which is slill agitating electrical engineers is that 

 of running more than one alternate-circuit dynamo machine 

 connected to the same system of mains. Before the matter 

 became one of practical concern, it was considered in this 

 room, and it was shown mathematically th.at it was possible to 

 ran independently-driven alternators in parallel hut impossible 

 to run them in series. That is to say, that if two alternators 

 were connected to the same mains they would tend to adjust 

 them<elves in relation to each other so that their currents coull 

 be added, but that if an attempt were made to couple them, so 

 that their pressures should be added, they would adjust them- 

 selves so that their effects would be opiiosed." 



Perhaps of all engineering problems which have received 

 their solution in the Last hundred years that of the greatest 

 practical importance is the c nvcrsion of the energy o( heat into 

 the enerity of visible mechanical motion. The science of 

 thermodynamics has advanced along with the practical improve- 

 ment of the steam-engine. By its aid, particularly by the aid 

 of ihe <o called second law, we know what is possible of attain- 

 ment by the engineer under given conditions of temperature. I 

 mu»t not trench on the subject of one of my successors, but I 

 may point out that our knowledge of the second law of thermo- 

 dynamic* wai first developed by means of maihematics, and 

 that today its neatest exprosion is by means of partial differen- 

 tial coeth. irn'.. The two mo«t notable names in cnnncciion 

 with the <levelopment of Ihe second law of ihermoilynamics in 

 harmony wiih the first are those of Krivin ami Clausius ; Imih 

 dealt with the subject in a mathematical form not compre- 



1 Ins'iiui' .n '.f \.\"\n<-»\ Kniintcri. NoTrmbcr tj, >B84. 



2 Minuics of lr,<,{r,limfi Init. C.E., April s, i8£3 ; 

 Electrical Engineers, N-ive>iit>rr ij, 1^84. 



NO. I 280, VOL. 50J 



lnililuli'>n of 



I henstble to those who have not had substantial mathematical 

 ! training. 



Illustrations such as these might be multiplied almost indefi- 

 niiely. They show that the advancement of the science of 

 I engineering has been aided in no inconsiderable measure by the 

 j labours of mathematicians directly applying the higher mathe- 

 I matical methods to engineering problems. They show, too, one 

 way in which respect lor a foruiula may be dangerous, one way in 

 which it i> true that mathematics may be a bad master. In St. 

 Venani's problems we have an example in which the use of older 

 results of limited application in cases where the assumptions on 

 which they rest are not true will mislead. The examples show 

 the proper remedy ; it is a more complete application of mathe- 

 matical methods. The error is just one which a man will make 

 who has the power to use a formula without a re.idy understand- 

 ing of how it is arrived at. A practical man, ignoring mathe- 

 matical results, might or might not escape the error of supposing 

 that a triangular shaft woula break at the angles under torsion 

 the halleducaied mathematician would certainly fill into llu 

 snare from which complete mathematical knowledge would 

 deliver him. You can only secure the services of ihat good 

 servant, maihematics, and escape the tyranny of a bad m.asler 

 by thoroughly mastering the branches of mathematics you use. 

 The mistake caused by the wrong application of mathematical 

 lormula; is only to be cured by a more abundant supply of mote 

 powerful mathematics. 



There is another drawback to the use of results, taken, it m.ay 

 be, out of an engineering pocket-b 10k by those who are not pre- 

 pared to understand how they are reached and on what founda- 

 tions they rest. The educational advantage is lost. The close 

 observation which enabled the earlier engineers to proportion 

 their means to the ends to be attained was no doubt very 

 laborious, and the results could not be applied to cases much 

 different from those which had tieen previously seen, but the 

 effect on the character of the engineer was great. In like 

 manner, to thoroughly understand the theory 01 an engineering 

 I>roblem makes a man able to understand other protilems, and 

 in addition to this precisely the same mathematical reasoning 

 applies to many cases. The mere unintelligent use ot a formula 

 loses all this ; it leaves the mind of the user unimproved, and it 

 gives no help in dealing with questions similar in form though 

 different in substance. 



But even the use of maihematics by competent mathemati- 

 cians is nut without drawbacks. Mathematical treatment of 

 any problem is always analytical — .inalyiical, 1 mean, in this 

 sense that attention is concentraied on certain facts, and other 

 facts are neglected for the moment. Kor exampi , in dealing 

 with the thermodynamics o( a steam-engine, one dismisses from 

 consideration very vital points esseniial to the successful work- 

 ing of the engine, questions of strength of pans, lubrication, 

 convenience for repairs. But if an engineer is 10 succeed he 

 must not fail to consider every element necessary to .success ; he 

 must have a practical instinct which wdl tell him whether the 

 instrument as a whole will succeed. 11 is mind must not be only 

 analytical, or he will be in danger of solving bits of the pro- 

 blems which his work presents, and of falling into fatal mistakes 

 on points which he has omiitcil to consider, and which the 

 plainest, intelligent practical man would avoid almost without 

 knowing it. 



Again, the powers of the strongest mathematician being 

 limited, there is a constant temptation to fit the fac's to suit the 

 maihematics, and to assume that the conclu.sions will have 

 greater accuracy than the premises from which they are 

 dciluccd. This is a trouble one meets wiih in other applications 

 of mathematics to experimenial science. In order to make the 

 subject amenalile to treatment, one finds, for example, 111 llic 

 science of magnetism, that it is boldly assumed that ihc magnc 

 lisatioii of magnetisalile material Is proporlional to the magne 

 lislng force, and the ratio has a name given to it, and conclusions 

 arc drawn liom the atsum[>tion, but the tact is, no such 

 propoitionality exists, and all conclusions resultin,; from the 

 assumption are so far invalid. Wherever pos^iblf, iiiaihtiii.atical 

 deductions should be frequently vcriiied liy refcieiice 10 oDserva- 

 lion or experiment, for the very simple reason ttiai they are only 

 deductions, and the preiitises Iroin whicn the tie tuctiuns arc 

 made may be inaccurate or may be iiicoinpleie We must 

 always remember that we cannot get more out of the mathe- 

 matical mill than we put intu it, thiugh we may get 11 in a form 

 infinitely more useful for our purpose. 



Engineers uo doubt regard iheir profession ft 0111 vtry different 



