250 ON A DYNAMICAL TOP. 



The method by which M. Poinsot has rendered the theory more manageable, 

 is by the liberal introduction of "appropriate ideas," chiefly of a geometrical 

 character, most of which had been rendered familiar to mathematicians by the 

 writings of Monge, but which then first became illustrations of this branch of 

 dynamics. If any further progress is to be made in simplifying and arranging 

 the theory, it must be by the method which Poins6t has repeatedly pointed out 

 as the only one which can lead to a true knowledge of the subject, that of 

 proceeding from one distinct idea to another, instead of trusting to symbols and 

 equations. 



An important contribution to our stock of appropriate ideas and methods has 

 lately been made by Mr R. B. Hayward, in a paper, " On a Direct Method of 

 estimating Velocities, Accelerations, and all similar quantities, with respect to axes, 

 moveable in any manner in Space." (Trans. Cambridge Phil. Soc. Vol. x. Part I.) 



* In this communication I intend to confine myself to that part of the 

 subject which the top is intended to illustrate, namely, the alteration of the 

 position of the axis in a body rotating freely about its centre of gravity. I 

 shall, therefore, deduce the theory as briefly as possible, from two considera- 

 tions only, the permanence of the original angular momentum in direction and 

 magnitude, and the permanence of the original vis viva. 



The mathematical difficulties of the theory of rotation arise chiefly from 

 the want of geometrical illustrations and sensible images, by which we might 

 fix the results of analysis in our minds. 



It is easy to understand the motion of a body revolving about a fixed axle. 

 Every point in the body describes a circle about the axis, and returns to its 

 original position after each complete revolution. But if the axle itself be in 

 motion, the paths of the different points of the body will no longer be circular 

 or re-entrant. Even the velocity of rotation about the axis requires a careful 

 definition, and the proposition that, in all motion about a fixed point, there is 

 always one line of particles forming an instantaneous axis, is usually given in 

 the form of a very repulsive mass of calculation. Most of these difficulties may 

 be got rid of by devoting a little attention to the mechanics and geometry of 

 the problem before entering on the discussion of the equations. 



Mr Hayward, in his paper already referred to, has made great use of the 

 mechanical conception of Angular Momentum. 



* 7th May, 1857. The paragraphs marked thus have 1*en rewritten since the paper was 



