8o 



NATURE 



{Nov. 2 2, 1888 



Except for an unessential disturbance due to friction, the pres- 

 sure all through tJie tube is uniform so far as the wind-motion is 

 concerned. 



Erect across the tube a couple of imaginary partitions, and 

 watch the substance streaming past them. The state of the 

 substance at either partition, whatever it may be at one instant, 

 remains permanently the same always ; hence the mass of sub- 

 stance inclosed between the two must remain permanently the 

 same — for it cannot be always steadily increasing — and therefore 

 the mass of matter flowing through any one plane is constant. 



In future attend to one of the planes only, and call the density 

 of the substance at this place p. The plane may be at or near a 

 condensation, it may be at or near a rarefaction, or, again, it may be 

 where the substance has its ordinary density ; whatever the state 

 of the substance there, the same it remains. The longitudinal- 

 pulse motion of the particles of substance (which has previously 

 been illustrated and discussed at length) is superposed upon the 

 wind-motion ; and if we followed any one particle along the 

 stream we should see it simply oscillating with a simple har- 

 monic motion — 



X = a sin nt, or v = an cos nt, 



having at any instant the velocity v. 



But we are not going to follow a single particle down stream ; 

 we are contemplating a procession of particles as they succes- 

 sively pass the fixed partition, and at the instant of passage they 

 are all in a definite phase of their motion — they all have the j 

 same definite velocity, v, as they pass, in addition to their general 

 wind-velocity, U. The vibratory velocity v may be in the same 

 direction as U, or it may be in the opposite direction ; it may 

 have any value between ± 11a, of course. So the resultant 

 velocity of each particle as it passes the fixed partition is 

 algebraically U 4- v. This represents the length of cylinder of 

 substance passing through the partition per second ; and, since 

 the partition is of unit area, the mass of substance flowing past 

 it per second is 



;w = (U -F z/)p, and is constant . . . . (i) 



This of itself is an interesting result ; for it shows that at the 

 middle of condensations, where p is a maximum, v must have its 

 greatest negative value ; and the particles are therefore all in full 

 swing back against the wind [i.e. travelling with the sound-pulse) 

 at the middle of every condensation. At a rarefaction, v has 

 its greatest positive value, and the particles are swinging with 

 the wind (against the sound-pulse). Only at half-way places, 

 where the density of the substance has its average or undisturbed 

 value, are the particles quiescent as regards the sound-pulse. 



Next consider the dynamics of the matter, and the force which 

 must act to vary the motion of the particles. 



If the pressure were the same on either side the partition, there 

 could be no change of velocity for the particles as they pass. 

 The charge of velocity, d{\] + v), or dv, must be due to 

 a difference of pressure existing on either side the partition ; 



and if the slope of pressure -f- is positive, the pressure is 



ax 

 greater on the lee-side of the partition than on the windward 



side, ami so the acceleration — will be negative. Hence, 



dt ^ 



equating the force acting and the momentum generated by it per 

 second, — 



dp — - nidv (2) 



This equation, along with equation (i), solves the problem, and 

 determines the velocity U. 

 Differentiating (i) — 



(U -f ii)d(, -H pdv = O, 

 Rewriting (2) by help of (i) — 



(U -f v)fdv = -- dp. 

 Substituting for pav from one of these into the other, we get — 



Hence we learn that the value of U is determined by calcu- 

 lating the ordinary value of -* for the medium in its uncom- 



dp 

 pressed and unrarefied state. 



So— 



\] = /dp 

 \ dp 



(4) 



This is the velocity with which the substance must flow 

 through the tube in order to keep the sound-pulse stationary : 

 this, therefore, is the velocity of sound in it. 



(U + vf = 

 dp 



dp 



dp 



(3) 



This equation shows that ^ is by no means constant all 

 dp 

 through the substance. It is greatest wherever v has its maximum 

 positive value — that is, at the centre of every condensation ; it is 

 least at the centre of every rarefaction ; it has an a\erage value in 

 the undisturbed portions of the medium, and it is there equal to U^. 



The result is general, and applies to all substances. But for 

 gases it may be written more explicitly by help of their charac- 

 teristic equation i- — RT, and their adiabatic condition/ « py -. 



viz. — 



U = V7l<T (5) 



T here means the undisturbed temperature of the gas, but if 

 one chooses to allow the equation to follow the fluctuations of 

 temperature (± /) adiabatically produced in the condensations and 

 rarefactions, one must write the more general form — 



(U -f vf = /C'R(T ±t) (6) 



which gives us the relation between fluctuation of temperature 

 and vibrational velocity. We may also write it — 



p-(T ± /) ^ const (7) 



which shows the connection between the elevation or depression 

 of temperature, and the density, at any part of a sound-wave. 



Speaking as a teacher, I believe one reason why we fail to 

 make things clear is, because we are often in too big a hurry. 

 One's natural tendency is to give such an investigation — as this, 

 for instance, in Maxwell's " Heat" — in a few lines on the black- 

 board, taking perhaps halfan hour or less overit, and forgetting that 

 it embodies m concentrated form a great deal of difficult thought, 

 though the actual mathematics may be simple. Gradually I am 

 learning not thus to scamper over che ground, but to lead up to 

 a thing in two or three or even more lectures, and then to devote 

 a whole hour to the thing itself. By this means, students may 

 ultimately be got to grip and feel the thing as a whole, instead 

 of having to ascend step by step to it ; but it is hopeless to expect 

 them to thus grasp it straight off; and even if it were possible, 

 it would not be really desirable for various reasons. The attempt 

 to hurry them into the comprehension of difficulties leads them, 

 I believe, into a vague notion that everything is hazy and half 

 unintelligible. The best thing we can do for them is to get them 

 to see some few things luminously, so that tney may not feel 

 inclined to rest satisfied with half-knowledge in other instances. 



Olivek J. Lodge. 



November 12. 



P. S. — Since writing the above, I have referred to Prof. 

 Everett's note A in Deschanel, and have found it excellent, like 

 all his notes ; he happens to have employed just the same means 

 as the above for obtaining equation (i), but for the latter part I 

 prefer my statement. I trust no one will imagine that the above 

 contains anything more than a way of putting things to students. 



The slip of a wrong sign in Maxwell I had not distinctly 

 noticed, but the simplest statement of it seems to be that in 

 obtaining the second equation he has put, for the change of 

 velocity of each particle as it passes a plane, du instead of 

 d{\} - u) ; that is, the change of absolute instead of relative 

 velocity. — O. J. L. 



A Simple Dynamo. 



I VENTURE to send you a brief description of a simple electro- 

 magnetic instrument which I have recently devised for illustrating 

 the principle of the Gramme ring. 



Two pulleys. A, B, having semicircular grooves, are mounted, as 

 shown in the figure, on a piece of board ; round the two wheels is 

 stretched a continuous coil of copper wire ; a horseshoe magnet is 

 placed with its poles close to the vertical parts of the coil ; the 

 wheels are connected to the terminals 1 1' : when the wheel A is 

 rotated the whole coil moves, and a steady current is at once 

 generated, which flows from terminal to terminal when they are 



