February 25, 19 15] 



NATURE 



715 



I will make one or two of the experiments with the 

 original gyrostats, but it will save time if I repeat the 

 others with some of the new and improved gyrostats 

 invented by Dr. J. G. Gray, whom I am fortunate 

 in having to assist me on the present occasion. 



[The usual experiments, illustrating precessional 

 motion of a gyrostat with the axis horizontal, while 

 under the influence of a couple due to the gyrostat 

 overhanging, or to a weight hung on one end of the 

 case surrounding the axle (Fig. 6) were performed.] 



This behaviour of the gyrostat is often considered 

 paradoxical, and must, I suppose, be regarded as 

 difficult to explain in a popular manner. At any rate, 

 the popular explanations are as a rule extremely un- 

 satisfactory. Yet in this particular case of horizon- 

 tality of the axis the matter is simple enough, I think. 

 Let me illustrate by means of this pedestal top 

 (Fig. 7). The curved arrowhead shows the direc- 

 tion of rotation, the projecting arrow the axis of 

 spin, the arrow pointing down can be turned so as 

 to show the direction of the axis of any applied 

 couple. First observe that when I try to retard the 

 precessional motion the axis descends, if I try to 

 accelerate the precession the axis rises. This experi- 

 ment shows that the horizontality of the axis depend; 



Fig. 6. 



Fig. 7. — Motor-Gyrostai in " Foik 

 and Pedestal " Mounting. 



on the freedom of the gyrostat to precess at a certain 

 definite rate. This rate, as we shall see presently, 

 depends on the couple applied by the weight of the 

 gyrostat acting downward in one vertical line, and 

 the pull of the string acting upward in another line 

 nearly vertical, and on the angular momentum of 

 the flywheel. 



Look at the thing in this way. The axis of 

 rotation round which the flywheel has angular mo- 

 mentum is turning as you see towards the horizontal 

 axis A of the couple, with angular speed, Q say. 

 Now, and this is the point not recognised as a rule, 

 this motion itself creates a rate of production of 

 angular momentum about the axis A of the couple. 

 For when an axis with which is associated a directed 

 quantity, L say, is turning towards a fixed direction 

 at right angles to it with angular speed C> there is 

 a time-rate of production of the quantity associated 

 with the latter direction measured by the product LQ. 



Now the flywheel is revolving with angular speed 

 w, so that if its moment of inertia is m fe-, 

 it has angular momentum m k'm about the 

 axis R; but with angular speed Q the axis R is 



turning towards the instantaneous position of the 

 axis A, a fixed direction to which R is at the moment 

 perpendicular, and, in consequence of this turning, a 

 rate of production of angular momentum m k'm . O 

 exists about A. 



Now for the steady motion of the gyrostat, that is, 

 steady turning in azimuth without rising or falling 

 of the axis, it is only necessary' that this rate should 

 be equal to the moment of the couple about A, G 

 let us sav. Thus we get mfe'ti>Q = G, which gives 



If I hurr>- the precession by giving a little impulse,, 

 and then leave the gyrostat to itself, the hurried 

 motion, if it continued afterwards in the horizontal. 

 plane, would result in a more rapid generation of 

 angular momentum about A than there is moment of 

 couple to account for, and the gyrostat would begin 

 to turn about A, in the direction to cause the angular 

 momentum to be produced at the proper rate, that is 

 the axis would begin to rise. In the same way an 

 impulse towards delaying the precession would cause 

 the axis to begin to descend. In each case the result 

 would be a succession of alternate rises and descents; 

 but the subject of vibrations about steady motion 

 will be found treated in the Appendix, § (5) [see 

 Journal, I.E.E.]. 



Here it is important to remark that there are two 

 possible precessional motions for the same spin and 

 the same inclination of the axis of spin to the vertical, 

 which are given in the theorj' as the roots of a certain 

 quadratic equation (see Appendix). One is great, the 

 other small. The former to the first approximation 

 does not depend on applied forces, the other does. 

 Lord Kelvin called the lormer "adynamic," the other 

 "precessional." But in strictness both involve the 

 forces, and they appear as the roots of a certain equa- 

 tion. One of these is at once approximately realised 

 when the wheel is spun fast, the g>TOStat set on the 

 plate at rest, and left to itself. The motion is one of 

 small oscillation about the steady motion, which is 

 characterised by slow precession, given very nearly, 

 but not quite exactly, by the same formula as before. 

 The other motion of the axis in the same cone is one 

 of much greater precessional angular speed. The 

 popular expositions which I have seen of g\TOStatic 

 steady motion as a rule ignore this second possible 

 motion. It can be realised by proper means. 



In strictness we must regard this second preces- 

 sional motion as characteristic also of the g>-rostat 

 when its axis is horizont.nl, but in that case the pre- 

 cessional angular speed is infinite, and only the slow 

 motion is realisable. 



The rule, often stated, that hurr>ing a gyrostat 

 in its precession causes tilting up of the axis, and 

 delaying the precession causes tilting downward, is 

 true only of the slower more usual precession. For 

 the faster precession exactly the reverse rule holds 

 good. This fact does not seem to be generally 

 known, as the rule is generally stated absolutely. 



It is important to notice that if the centre of 

 gravity of the gyrostat is above the point of sup- 

 port, supposed on the line of the axis, the two pre- 

 cessional motions are in the same direction ; if, on 

 the other hand, the centre of gravity be below the 

 point of support, the precessional motions are^ in 

 opposite directions. The faster motion changes sign 

 in passing through an infinite value, when the axis 

 is horizontal. 



By the effect of hurrying or retarding the precession 

 was sometimes explained in our lectures the rising 

 and fallinfr of a top soinnine on a rounded peg in 

 contact with a rough floor along which the top c.nn 

 move. At first the spin is fast and the slioping is^ 

 such as to produce a hurrying friction couple which 

 causes the erection of the top. .\fter the spin has 



