Rotatory Motion, the Gyroscope, 4'C. 131 



revolve on another in the same plane with the first, it will revolve on 

 neither, but on a line dividing the angle which they contain, so that the 

 sines of the parts are in the inverse ratio of the angular velocities with 

 which the body would have revolved about the said axes separately."* 

 Airy, in his Mathematical Tracts, demonstrates this theorem, and a 

 variety of others connected with it, as a foundation for his elaborate inves- 

 tigation and quantitative calculation of the precession of the equinoxes. 



As Playfair says, " A body free to rotate about any axis, will not 

 rotate permanently about any particular axis, unless the centrifugal 

 forces are balanced with respect to that axis." Airy, however, says, 

 " Since the axis about which the earth is at any instant revolving, does 

 not coincide with the axis of the figure, the centrifugal force will 

 diminish the effect produced by the distant body. With an ellipticity, 

 however, so small as that of the earth, this diminution is not sensible." 

 This seems to indicate that he thinks the earth does not rotate about 

 its axis of figure, but is continually rotating about a new line. Whether 

 this is so or not, I do not see how the centrifugal forces could in any 

 case diminish the action to which he refers — namely, that of the sun 

 and moon — by which the precession of the equinoxes is occasioned ; for 

 the dii'ections of the centrifugal forces are at right angles to, and cannot 

 affect the continually changing position in space which the axis should 

 assume, whilst they must continue to act on the earth as long as the 

 axis of figure does not coincide with this position. Again, if with an 

 increased ellipticity of the earth, the centrifugal forces were, according 

 to Airy's theory, to diminish the precessional effect of the distant body, 

 the inchuation of the equator to the ecliptic would be gradually reduced 

 by the action of the sun and moon.f I may here remark, that this 



* This enunciation of the theorem should be strictly adhered to. In the statement of 

 it to be found in some modern works there is a vagueness which ignores a distinction that 

 exists between the " composition of rectilinear motion," and the " composition of rotatory 

 motion." Wlien two rectilinear motions are compounded, the effect, as regards the 

 position of the body acted upon, is the same as if each took place separately, one after the 

 other; but it is not so in the case of rotatory motions. It is scarcely accurate to say, 

 without some qualification, that two rotations can be replaced by a single rotation, having 

 a relation to the two rotations, analogous to that which the diagonal of a parallelogram 

 ha3 to the sides. We can only conceive of two rotations as acting at separate times, and 

 a particle submitted to them in succession will not be finally in the position in which the 

 single rotation compounded of the two would place if. 



t It will be seen from what is said farther on that in some cases the precessional motion 

 U undulatory. If, in the case of tlie earth, the centrifugal resultant has been insulHcient 

 to prevent the undulations at once, it would still do so in time if it had any existence at 

 all, by the accumulation of small elTects ; or, as Poisson shows, a determinate impulse 

 in the direction of llie processional motion would prevent the imdiilutions. It is not tlio 

 niilatidii that I litre rcfir to. 



