1866.] 



moving freely about a Fixed Point, 



141 



where X is an arbitrary constant. If, now, we revert to the natural supposi- 

 tion of the top being of uniform density, it is well known that 



A : B : C : : fi' + c" : c- + «' : a--\-h\ 

 and these ratios may be identified wdth those above (although this would 

 not at the first blush be supposed to be the case) by giving a suitable value 

 to the arbitrary constant \. 



Thus, then, Poinsot's central ellipsoid supposed of uniform density and 

 set spinning upon a roughened invariable plane will represent the motion 

 of a free rotating solid, not in space only but also in time ; the body and 

 the top may be conceived as continually moving round the same axis and 

 at the same rate at each moment of time*. 



The problem of the top is completed in the memoir by applying the 

 general Eulerian equations to determine the friction and pressure, a process 

 which involves some rather operose but successfully executed algebraical 

 calculations. 



I next proceed to account analytically for the kinematical theory esta- 

 blished at the outset of the memoir, and in doing so am necessarily led to 

 give greater completeness to it, and at the same time an extension to the 

 existing theory of confocal surfaces of the second order, by introducing the 

 complementary motion of surfaces that I call contrafocal to one another : 

 confocal ellipsoids are those the differences between the squares of whose 

 corresponding principal arcs are all three the same ; contrafocal elhpsoids 

 I define to be those the sums of the squares of whose corresponding arcs 

 are the same. Any two bodies whose central ellipsoids are either confocal 

 or contrafocal I term related — correlated in the one case, contrarelated in 

 the other, and I show that the kinematical construction in question is only 

 another rendering of the first of the propositions herein subjoined concern- 

 ing bodies so related. 



1st. If two correlated bodies be placed with their principal axes respec- 

 tively parallel and be set spinning by the same impulsive couple, they 

 will move so that the corresponding axes of the one and the other body 

 will continue always equally inclined to the axis of the couple, and their 

 original paralleHsm at any instant may be restored by turning one of the 

 bodies about this last-named axis through an angle proportional to the time 

 elapsed since the commencement of the motion. Virtually, this amounts 

 to saying that the difference between the displacement of two correlated 

 bodies subject to the same initial impulse is equivalent to a simple uniform 

 motion about the invariable line. 



2nd. So, in like manner, if the bodies be contrarelated, the sum of their 

 displacements is equivalent to a simple uniform motion about such line. 



* Accordingly, if we conceive any body as lying wholly in the interior of the ellipsoidal 

 top, which is its kinematical exponent, such body will move precisely as if it were free, 

 and consequently its density may be uniformly increased or diminished in any ratio, or it 

 may be entirely removed without affecting the law of the motion of the surrounding 

 crust in relation to space ur time. 



