1866.] 



moving freely about a Fixed Point. 



143 



cone") that in the problem of the motion of a free body, of whatever form 

 and subject to whatever initial conditions one pleases, there enter but two 

 arbitrary parameters. Calling A, B, A + B the three moments of inertia 

 of one or the other equivalent disks, L the magnitude of the impulsive 



couple, M the vis viva, these two parameters (say p, q) will be ^ ; if 

 to them we add a quantity — » say r (where t is the terms reckoned, as it 



may be, from an intrinsic epoch as explained in the memoir), all other ele- 

 ments of the motion, as the total or partial velocities and the angles, what- 

 ever they may be, selected to determine the position of the rotating body 

 become known functions of these three quantities, q, t, and may be re- 

 duced to tables of triple entry, or be graphically represented by a few 

 charts of curves ; and it should be noticed that p, the smaller of the two 

 parameters^, q, will be always necessarily included between o and 1, and 

 that the other parameter q may, by a due choice of the species of reduction 

 adopted, be forcibly retained within the same limits. The five quantities 



py q, \, p-\-q will then form an ascending series of magnitudes subject 

 only to the liability of the middle term q to become equal to 1 on the one 

 hand, or to p on the other : q becoming unity corresponds to the case of the 

 so-called "Dividing Polhode," Legendre's 2nd case cas tres re- 

 marquable'') ; and q becoming equal to p is of course the case of the body 

 itself, or its "central ellipsoid," becoming a figure of revolution, in which 

 case the motion is practically the same as that of a uniform circular plate. 



Besides these two exceptional cases, the only singular cases properly so 

 called, the quinary scale of magnitudes just exhibited serves to indicate all 

 the more remarkable cases (requiring or inviting particular methods of 

 treatment) which can present themselves in the theory. These may be dis- 

 tinguished into special cases, which arise from any two consecutive terms 

 becoming (to use Prof. De Morgan's expressive term) subequal, i. e. dif- 

 fering from one another by a quantity whose square may be neglected, and 

 double special cases, which arise when any three consecutive terms become 

 subequal ; all of which, together with peculiar subcases appertaining to the 

 double special class, perhaps deserve more thorough examination than may 

 have been hitherto accorded to them. I conclude the memoir with pointing 

 out the place which this problem of Rotation appears to me to occupy in 

 dynamical theory, as belonging to a natural and perfectly well defined group 

 of questions, of which the motion of a body attracted to two fixed centres and 

 the renowned problem of three bodies acted on by their mutual attractions 

 are conspicuous instances. This group is characterized by the feature 

 that, as regards them, equations of motion admit of being constructed, from 

 which not only the element of time, as in ordinary mechanical problems, but 

 also an element of absolute space is shut out ; supposing the equations thus 

 reduced by two in the number of the variables to have been integrated, 

 Jacobi's theory of the last multiplier serves to reduce both the excluded 



