180 TENTH ANNUAL REPORT OF 



. Now, these nodal motions in the solar system, of which I hare 

 named the two most familiar examples, are the effects of some diatm'b- 

 ing force ; for we have seen that, without disturbance, the plane of 

 rotation would he forever parallel to itself, and woiild therefore cut a 

 fixed plane always in the same points. I have already alluded to the 

 law of composition in rectilinear motions ; namely, that the resultant 

 motion lies between the directions of the two component forces, divi- 

 ding the angle into two parts, which have a very simple relation to the 

 magnitude of the forces, the body moving most nearly in the direction 

 of the greater force. The law of composition of rotary^ motions is 

 quite analogous to it, and directly deducible from it. It is this : If a 

 body is revolving on an axis, and a force is applied tending to revolve 

 it on some other axis, it will not revolve on either, but on a third one^ 

 between the two, and dividing the angle as before.* 



To show you the truth of this law, I whirl the spheroid of the rota- 

 scope, so that, while the south end of the axis points from_ me, the 

 particles pass over from my left to my right. Now, with this smooth 

 rod, I press down the north side of the inner ring, thus tending to 

 give the spheroid a similar right-hand rotation on an 

 axis pointing westward. The effect is, you perceive^ 

 that the ring slips round under the rod, so as to 

 bring the south end of the axis into the southwest 

 quarter — that is, behveen the two axes of separate ro- 

 tation. If I continue the pressure, the axis passes 

 round still farther west, endeavoring each moment to 

 place itself between its present position and one at 

 right angles to itself f If there were no friction un- 

 der the rod and on the pivots, this horizontal rota- 

 tion would continue so long as the pressure is ap- 

 plied, and more rapidly as the pressure is greater. But, as there is 

 friction, the south end of the axis slowly rises from a horizontal plane. 

 I now direct the axis again towards the south, and press the north side 

 of the ring upivard — that is, I endeavor to produce a right-handed rota- 

 tion on an axis pointing eashvard ; and you see the south pole imme- 

 diately pass round towards the east, between the two axes. 



As all the cases of compound rotation are more easily described by 



="= I did not think it best, in a popular lecture, to give a full and technical statement of 

 the laws of composition, in either rectilinear or rot 'ry motions. They are suhjoined here 

 for the use of anv Avho may wish to recur to them : 



"If a particle "receives two motions, which are separately represented by the adjacent 

 sides of a parallelogram, the resultant motion is represented by the diagonal of the same ; 

 and therefore, in direction, it divides the angle of the components, so that the sines of the 

 two parts are inversely as the components ; and in quantity, it has to either component the 

 same ratio as the sine "of the whole angle has to the sine of the part between itself and 

 the o//(er component." 



The law of compound revolutions is this : "If a body receives two impulses, one of whicti 

 would cause it to revolve on <mc axis, and the other on a second, it will revolve on a third 

 axis, situated between the two, and dividing their angle, so that the sines of the parts are 

 inversely as the two impulses. And the velocity of rt)tation is to the velocity due to eiiliei' 

 ImpulKC, as the sine of the angle between the two original axes is to the sine of the partial 

 angle between the third axis and that on which the other impulse would have revolved 

 the body." ^ 



t In figure 5, the particles at A, moving in the direction of the arrow by the revolu- 

 tion of the spheroid, and also urged towards the rod, by which the ring is pressed down, 

 move lelween these two directions ; this is effected by tlie sliding of the ring towards the 

 left, under the rod, as shown by the dmbk-ihaft arrow. 



