450 



NATURE 



[March 12, 1908 



constant angular velocity o, exerting the moment ma. upon 

 tlie shelf D, and therefore on the car.' 



This precession continues until the roller and the shelf 

 cease to touch. At first H lifts with the shelf, and after- 

 wards the shelf moves downwards, followed for some 

 <listance by the roller. If the tilt had been in the opposite 

 direction, the shelf D would have acted upon the roller 

 H', and caused just the opposite kind of precession, and 

 a moment of the opposite kind. 



We now have the spindles out of their mid-positions as 

 00, O'Q'. How are they brought back to NOO'N' with 

 H permanently lowered? 



It is the essence of Mr. Brennan's invention that, after 

 a restoring moment has been applied to the car, the 

 spindles shall go back to the position NOO'N' with H 

 permanently lowered, so as to be ready to act again. 



He effects this object in various ways. Some ways 

 described in his patents are quite different from what is 

 used on the model, and the method to be used on the full- 

 size waggon will again be quite different. I will describe 

 one of these methods. Mr. Brennan tells me that he 

 considers this old method to be crude, but he is naturally 

 unwilling to allow me to publish his latest method. 



D' is a circular shelf extending from the mid-position 

 in my direction ; D is a similar shelf extending from the 

 mid-position into the paper or away from mc. It is on 

 these shelves that H' and H roll, causing precession, as 

 I have just described. When H' is inside the paper or 

 when H is outside the paper they find no shelf to roll 

 upon. There are, however, two other shelves, L and L', 

 for two other rollers, M and M', which are attached to 

 the frames concentric with the spindles. They are free to 

 rotate, but are not rotated by the spindles. When they 

 are pressed by their shelves L or L' this causes negative 

 precession, and fhey roll towards the NOO'N' position. 

 There is, of course, friction at their supports retarding 

 their rotation, and therefore the precession. The important 

 thing to remember is that H and H' when they touch 

 their shelves (when one is touching, the other is not touch- 

 ing) cause a precession away from the mid-position at a 

 rate a which produces a restoring moment ma. of constant 

 amount (except for slipping), whereas when M or M' 

 touches its shelf L or L' (when one is touching the other 

 is not touching), the pressure on the shelf and friction 

 determine the rate of precession towards the mid-position 

 as well as the small vertical motion. 



Suppose the tilt to be corrected is R, when D presses 

 H upward. The moment ma. and its time of action (the 

 total momenta! impulse) are too great, and R is over- 

 corrected ; this causes the rollers M' to act on L', and the 

 spindles return to the mid-position ; they go beyond the 

 mid-position, and now the roller H' acts on D', and there 

 IS a return to the mid-position and beyond it, and so it 

 goes on — the swings of the gyrostats out of and into the 

 mid-position, and the vibrations of the car about its position 

 of equilibrium getting rapidly less and less, until again 

 neither H or H' nor M or M' is touching a shelf. It 

 is indeed marvellous to see how rapidly the swings decay." 



It will be seen that by using the two gyrostats instead 

 of one, when there is a curve on the line, although the 

 plane NOO'N' rotates, and we may say that the gyrostats 

 precess, the tilting couples which they exercise are equal 

 and opposite. 



It is evident that this method of Mr. Brennan is 

 altogether different in character from that of Mr. Schlick. 

 1 I am ';upDos'n5 the prece«ionala'ieIes to be ^mall ; whci th^ alleles are 

 like NOQ,NO'Q'. Fig. 3, the sum of the moments of the two gyrostats 

 would be mo. cos NOQ, a b^ing constant, if there were no slipping ; but 

 there is always slipping. anH the eo->d workinc; nf the^ anparatns requires 

 that there shall be such slipping, a is not constant, and it is always less than 

 what it would be if there was true rolling. 



1 If in Fig. ^ R is the angle which the waggon makes with its position of 

 equilibrium ; if M is the moment with which the shelf D acts clockwise 

 upon H, and P is the angle of precession QoV ; and if » is the coefficient 

 of friction between D and H, th»n in the first part of the action above 

 described IS^R +weP - M =0. >«9R - /»=P+(»M = o, diS^- /ra.)'! -l-M=o. if 

 II, 1 and U are moments of inertia "f the wagg'^n about the rail, of the 

 frames about C. and of the frame G about F.J ; Am is the moment of 

 momentum of either wheel. These equations are easily solved on the con- 

 ditions that at t = a. ■R = R|,, eR=o, P=o, WP=o. .-Assuming no play, tfiat 

 s, that as soon as H leaves D. M' touches V, we cin now find the return 

 to the mid-positi 'n from a new set of equations. Friction re'ards the return. 

 It must he remembered. The motions nre exceedingly interesting when 

 num-rical values of T,. I, /, Sc. are tiken. but the practical min will find it 

 more interesting to make an experimental study of what happens. 



NO. 2002, VOL. ']']'\ 



Work is here actually done which must be supplied by 

 the electromotors. The restoring moment appliecl to the 

 car may be made as great as we please by increasing the 

 diameter of H. It is true that we cannot in this way 

 alter the total momental iinpulse, and this is the important 

 thing. 



One of the most important things to know is this : 

 the Brennan model is wonderfully successful ; the weight 

 of the apparatus is not a large fraction of the weight of 

 the waggon ; will this also be the case with a real car 

 which weighs 1000 times as much? If at any instant a 

 condition of things is suddenly produced so that the 

 waggon makes an angle R„ with its position of equil- 

 ibriuiTi, if its weight is W, its centre of gravity at the 

 height h above the rail, if I, is its inoinent of inertia 

 about the rail as axis, if \m is the moment of momentum 

 of each gyrostat wheel, the momental impulse mP, ought, 

 roughly, to be equal to CR„\/I|\V/i, where C is a constant. 

 I use P, for the total angular precession at first. Now 

 the tilt to be corrected, R^, may be due to wind pressure, 

 to a sudden shifting of the centre of gravity, or to centri- 

 fugal force, and it is not easy to compare these things in 

 waggons of different sizes. If, however, we take it that 

 the size of each dimension of the waggon is multiplied by 

 H and the size of each dimension of the gyrostatic 

 apparatus is multiplied by />, and the wheels have the 

 same peripheral speeds, we find the following results : — 

 For wind, R„ is proportional to n-'. For centrifugal 

 force it seems reasonable to take the speed of a waggon 

 as proportional to 11, and mean radius of rail curves pro- 

 portional to n-. In this case the result is again that 

 R„ is proportional to n-'. As for a possible accidental 

 shifting of the centre of gravity because of the displace- 

 ment of part of the cargo w through the distance (f, if we 

 take 7t;o=ii- and Ac-.n, we find with greater and greater 

 accuracy as iy is a smaller fraction of the whole weight, 

 R„ccn-"\ 



Taking P,, the maximum angle of precession, to be the 

 same in all cases, the above relation leads to the result 

 that ^ = K , or that the fractional weight of the apparatus 

 as compared with that of the waggon is proportional to 

 W-"''-", where W is the weight of the waggon. Thus, if 

 we take n=io, that is, every dimension of the model 

 inultiplied by 10 or its weight by 1000, then ^ = 7-5. so 

 that the weight of the gyrostat apparatus is only multi- 

 plied by 420. If in the model the apparatus was 10 per 

 cent, of the whole weight, in the large waggon the 

 apparatus is only 4 per cent, of the whole weight. In 

 fact, the larger the waggon the less proportion of its 

 weight and volume is occupied by the anparatus, a result 

 which inust be very satisfactory to Mr. Brennan. 



In the cases both of Mr. Schlick and Mr. Brennan, it 

 has to be remembered that if the diameter of the wheel 

 be increased in greater proportion than the dimensions of 

 the ship or waggon, or other dimensions of the wheel, the 

 proportional weight of the apparatus may be diminished. 

 K wheel of twice the diameter, but of the same weight, 

 may have twice the moment of momenium and may be 

 twice as effective. I assume the stresses in the material 

 to be the same. 



OAT THE PU\S1C.\1. ASPECT OF THE .ATOMIC 



THEORY.' 

 "T^HE lecture began by setting out a physical reason 

 -'• a priori why matter should be constituted of discrete 

 particles instead of being continuous. The requirements 

 of physics demand an sether to serve as the means of com- 

 inunication between portions of matter out of contact with 

 each other, and space can hardly be conceived as fully 

 occupied simultaneously by two media, matter and jEther ; 

 hence the matter must be constituted of discrete centres, or 

 nuclei, determining permanent collocations of energy in 

 the sether, which arc, in fact, primordial atoms and their 

 fields of force. The feasible problein of atomic physics is 

 to build up an adequate idea of the dynamic constitution 

 of these a^thereal fields of force ; there is the problem 

 1 .\bstract of the Wilde Lecture of the Manc-he=ter Literary and Philo- 

 sophical Society, delivered on March 3 by Prof J. L:irmor, Sec.R.S. 



