44 



NA TURE 



[May lo, ib94 



the web. This problem St. Venant also deals with in a rigid 

 mtthematical manner. Amongst other things, he showed that 

 a girder of rectangular section, such as shown in Fig. 4, would, 

 when bent, take the form shown by the curved lines in the same 

 Fig. The last two examples show how a little knowledge may 

 be a dangerous thing and how easy it is for anyone who 

 attempts to apply mathematics without adequate mathematical 

 knowledge to he misled. 



The theory of thicl< cylinders under bursting strc!;s from within 

 has many important practical applications to hydraulic presses 

 and to guns. It has been discussed more than once in this room. 

 As usual in considering these cases we are immediately led to 

 differential equations which here are fortunately solved without 

 serious difficulty, and the solution tells us the whole story. We 

 learn that doubling the thickness by no means doubles the 

 strength of the cylinder. .-\nd as a converse, that doubling the 



ir 



JL 



Fu;. 3. 



strength of the material will permit the thickness to be dimin- 

 ished lo much less than one-half. Twenty-five years ago 

 hydraulic presses were mostly made of cast-iron. Many people 

 were not a little astonished at the great reduction in thickness 

 and weight which became possible when steel was substituted 

 for the weaker material. In the case of guns it is well known 

 that greater strength can be obtained if the outer hoops are 

 shrunk on to the inner ones. ' Mathematical theory tells us what 

 amount of shrinkage should give the best results. It may 

 possibly not be worth while lo follow the results of theory pre- 

 cisely, but without the guidance ..f theory it would not be un- 

 natural to give so great a shrinkage that the gun would be 

 weaker than if no shrinkage were used. 



The rollmg of ships in a seaway gives an illustration of a 

 principle which has very varied application in many branches of 

 physics. Suppose a body is capable of oscillating in a certain 

 periodic time, and that it is submitted to a disturbing force of 



Fic. 4. 



given period, the equation of molion easily shows that the re- 

 sulting disturbance » ill be great if the two periods are equal or 

 nearly equal. \Vc meet with the principle in acoustics as re- 

 sonance. If two tuning-foiks are tuned 10 the same pitch, and 

 one is sounded in the neighbourhood of the other, that other 

 will presently be thrown into vibration by the waves transmitted 

 through the air from ihc firit. Vou may try a similar experi- 

 ment at any time on any piano. Strike the higher G in the 

 treble, the sound ceases on raising the finger. Now hold down 

 the middle C, and again strike (j ; the C siring at once takes 

 up the note sounded, and can be heard after the exciting string 

 has l)cen silenced by damping. The same fundamental idea is 

 found in the lunar theory in the term in the equation known as 

 the cvcction, and again in the theory of Jupiter's satellites. 

 The rc.is'in why the metals present in the solar atmosphere give 

 black lines in the spectrum l.y absorption, corresponding in 

 position with the bright line.i in ihc spectrum which the same 

 metals give when incandeiccnl, is again the same. Gas will absorb 



' VMXBi, " \^ta\ xra la Th^orie Math<malique ile I'Kluilcil^ des corf.'; 

 tolidcs." ulh \jt\tifu 



NO. labo, VOL. 50] 



or lake in from the ether waves of the ex.-ict period which it is 

 capable of giving to the ether. The general explanation of all 

 these phenomena is easy. Imagine a pendulum, and suppose 

 it experiences a periodic disturbing force, the first impulse of 

 the disturbing force gives the pendulum a slight swing ; the 

 effect of the second impulse depends entirely on when it occurs ; 

 it may occur so as to neutralise the eflect of the first, or it may 

 occur so as to increase it. If the period of the force is the same 

 as the natural period of the pendulum, the eflect of the second, 

 third, and later impulses will be added to the eflect of the first, 

 and the final disturbance will be great, even though the indi- 

 vidual impulses be minute. But the mathematical theory tells 

 us much more than any general explanation can do. It tells us 

 exactly what the character of the elTect will be, and its amount 

 if the periods are nearly but not exactly the same. It tells us, 

 too, exactly how friction aftects the results. .\nd the beauty of 

 it is that the mathematical theory is much the same in all cases, 

 so that having learned to deal with one case we are enlightened 

 as to a host of others. The oscillating body may be an iron- 

 clad, or it may bean atom of hydrogen ; the disturbing periodic 

 force may be the waves of the .Atlantic, or it may be the waves 

 in the ether occurring five hundred millions of millions of times 

 in a second ; it is all one to the mathematician ; the treatment 

 is substantially the same. 



The question of the speed of ships and the power to propel 

 them is probably more efiectually treated by experiment on 

 models, as was done by the late Mr. Froude, than by mathe- 

 matics alone ; but in order to learn from the experiments all 

 they are capable of teaching, a mathematical understanding is 

 needed. Given that we know by experiment all about a given 

 model, that we know what force is needed lo propel it at every 

 speed, we want to know from these experiments how a great 

 ship, 100 times as big, but similar in form in every respect, will 

 behave ; and here mathematics come in to aid us in m.iking the 

 inference. 



The construction of ships at once leads us on to the methods 

 of navigating them. In navigation I should find much material 

 for my purpose, but navigation is not usually included in 

 engineering, but many of the implements of navigation un- 

 doubicdly arc. The mariners' compass has for ages been the 

 mainstay of the navigator, and a simple enough instrument it 

 was till it was disturbed by the iron of which ships came to be 

 built. The distur'oance of the compass by the iron of the ship 

 was first seriously attacked by two senior wranglers. Sir G. 

 Airy and Mr. Archibald Smith. The disturbance may be 

 divided into two parts, the first due lo the permanent mag- 

 netism of the ship, the second to the temporary m.ignetism 

 induced by the earth's inductive action on the iron of the ship — 

 the first causes the semicircular, the second thequidrantal, error. 

 One has only to open the "Admiralty Manual of Deviations 

 of the Compass" to see how the malhematics of Archibald 

 Smith have accomplished a proper understanding of the sub- 

 ject. The errors of the compass are deal! with in two ways : 

 they are compensated by soft iron correctors, and by permanent 

 magnets so placed as to have an effect equal and opposite to the 

 effect of the temporary and permanent m.ngnetism of the «hip. 

 Or they are dealt with by forunil.T of correction which enable 

 the error to be calculated when the course of the shi[> antl the 

 conditions of the earth's magnetism are given, or a combination 

 of the two methods is used. Either method is based on Archi- 

 bald Smith's theory. It is not possible lo leave the subject of 

 the manners' compass without referring to the great improve- 

 ments of Lord Kelvin. The improvements relate to every part 

 of the instrumeni, and I venture to say that none of them could 

 have been made by anyone but a mathematician. In order to 

 get his card steady he knew that its period must be diflerent to 

 any possible period of the waves, or he would have the reson- 

 ance to which I have just referred coming in, so he gave his 

 card a considerable moment of inertia ; but this was managed 

 with a light card so that small needles could be used. If the 

 needles arc small the correction by soft iron masses and by per- 

 manent magnets is easier and more accurate. Then the bowl 

 of the compass had to be suitably carried so that it would not 

 be unduly disturbed by shock, and provision had to be made 

 for damping by fluid friction the oscillalions of the bowl if they 

 occurred. Lastly, a most beautiful method of correcting the 

 compa.ss, without taking a sight, was discovered. In every 

 detailed improvement one can detect that the inventing mind 

 was that of a most able and trained mathematician. 



An essential of safe navigation is an efficient system of light- 



