142 



Prof. J. J. Sylvester on a Rigid Body [May 17, 



3rd. In either case alike, the difference between the squared angular 

 velocities of the related bodies is constant throughout the motion. 



From these propositions it follows that for all practical intents and pur- 

 poses the motion of any body is sufficiently represented by the motion of 

 any other one correlated or contrarelated to it. To a spectator on the in- 

 variable plane the apparent motion of one rotating body may be made 

 identical with that of any other related one by merely making the plane 

 on which he stands turn uniformly round a perpendicular axis. It becomes 

 natural, then, to ascertain whether there is not always some one or more 

 simplest form or forms which may be selected out of the whole couple of 

 infinite series of related bodies, which may conveniently be adopted as the 

 exemplar or type of all the rest. Obviousty, the best suited for such pur- 

 pose will be a body reduced to only two dimensions, in other words an 

 indefinitely flattened disk, provided that it be possible in all cases to find a 

 disk correlated or contrarelated to any given solid 



The algebraical investigation for ascertaining the existence of such disk 

 is the same whichever species of relation is made the subject of inquiry, and 

 leads to the construction of three quadratic equations corresponding to the 

 respective suppositions of the original body becoming indefinitely flattened 

 in the direction of each of its three principal axes in turn ; so that for a 

 moment it might be supposed that the number of disks fulfilling the required 

 condition could, according to circumstances, be zero, two, four, or six. 

 But on closer examination, and bearing in mind that negative equally with 

 imaginary moments of inertia are inadmissible, it turns out that there 

 are always two such disks, and no more (except in the case of two of the mo- 

 ments of inertia being equal when the solution becomes unique). Of these 

 two disks, one will be correlated and the other contrarelated to the given 

 body, and they will be respectively perpendicular to the axes of greatest and 

 least moments of inertia. We have thus the choice between two methods of 

 reduction to the type form, and this choice is not a matter of unimportance 

 (in nature nothing exists in vain) ; for by means thereof the motion of 

 any given body subject to any initial conditions can be made to depend 

 upon either at will of the two comprehensive cases (Legendre's 1st and 

 3rd) to which the motion of a free rotating body is usually referred, so 

 that the distinction between these two cases (corresponding to the two 

 species of Polhodes on either side of the "Dividing Polhode," according 

 to Poinsot's method of exposition) is virtually abrogated. 



From the preceding theory, it follows (as also may be made to appear 

 alike from an attentive synoptic view of the commonly received analytical 

 formulae as from Poinsot's theory of the associated " sliding and rolling 



* The peculiar feature in the absolute motion of a disk is, that whilst it is turning in 

 its own plane with a variable velocity, the rate at which it turns about itself is constant, 

 as will at once become evident from eliminating r between the two equations 



Ap- +B!?^ +(A+B)r- =M, 



A y- -f- B^^i/^-f ( A -f B) V = L^ 



