Mechanical Illustrations of the Planetary Motions, 319 



that, when such is the case, the top will rise towards a ver- 

 tical position. 



For the sake of simplicity, I have spoken of the distances 

 set off, and consequently the abscissae and ordinates as of 

 definite lengths ; but, to those who have made mathematical 

 subjects their study, it will be evident that, in order to be 

 strictly correct, the second point in each curve must be taken 

 in immediate succession to the first, making the first ordi- 

 nate and abscissa infinitely small. In this case their rela- 

 tive magnitudes may be calculated by means of the differential 

 calculus ; or the result may, I think, be shown to depend 

 upon the following principle, which, if what I have already 

 said be admitted,* will be a self-evident consequence of it. 

 When the radius of curvature of the ellipse, at its lowest 

 point, is greater than that of the parabolic curve at its ver- 

 tex, the top will fall ; when less, it will rise. 



The same theory applied to that form of the top in which 

 the centre of gravity is below the centre of motion will show 

 that the conical revolution of the axis must then be back- 

 ward, or in a contrary direction to that of the rotation ; for 

 in this case the tendency of the top, when at rest, is not to 

 fall but to attain a vertical position. The lowest point, B, 

 in the circumference, having a tendency to rise, and the 

 highest point, A, to fall, both 



these tendencies will, by means ^ i r ^"*-^w 

 of the rotation, produce their f ^\ 



effect in advance of the highest L '■ j 4 



and lowest points, depressing \^ J 



the point a and raising the point ^^^ j -jr^ 



b. If we stop the motion of the 



top and produce the same effect with the finger, we shall find 

 that the highest point is thus thrown back to a', and the lowest 



* There will probably be some hesitation in accepting the preceding part of 

 this investigation as strict demonstration. I have the same hesitation myself, 

 and rest nothing upon it. I rather throw it out as a suggestion for considera- 

 tion. It has at least simplicity in its favour, which Euler's theory assuredly 

 has not. My doubts, however, do not extend to the main point, — of the ten- 

 dency of the top to fall or rise being converted by the rotation into the for- 

 ward or backward precessional movement. That does not admit of doubt, and 

 is the only part of the theory which I use for astronomical application. 



y2 



