778 PEOFESSOR SYL^^STEK ON THE MOTION OF A RIGID BODY 



the angles given in the tables in lieu of the angles themselves. In the special case of 

 a body with two equal moments of inertia, were not the simplicity of the motion such 

 as to render tabulation unnecessary, a distinct set of tables of double entry would of 

 course be employed. It is, I think, conceivable that the supposed tables of treble 

 entry might be of some practical value in studying by arithmetical or graphical methods 

 the geological phenomenon of evagation of the pole of the earth regarded as a body of 

 irregular form, and in other dynamical problems of a gyroscopical character where an 

 exact determination of the effect of a given disturbing cause might be difficult or imat- 

 tainable. 



The fact that there are no essential differences in the motion of a rigid body of any 

 form and started under any initial circumstances whatever, but such as depend upon 

 the particular values of the two positive proper fractions j,, jj, enables us at once to see 

 what are the special cases which alone can arise, and whether or no there is any real 

 distinction to be made between the general cases of the theory. At first sight it would 

 seem that four essential parameters enter into the question, the ratios of the initial 

 values of the partial velocities a„ a;,, Wj, and the ratios of the constants A : B : C, the 

 principal moments of inertia ; but one parameter is saved by the substitution of an 

 indefinitely flattened disk for a solid, and another by the introduction of an intrinsic 

 epoch fi'om which the time is reckoned, and thus a table of treble instead of quintuple 

 entry is competent to represent every possible variety of conditions. 



The problem that has been treated of in the foregoing pages is one (and possibly 

 the simplest) instance of a well-defined class of dynamical questions subject to a 

 peculiar method of treatment, which consists in the postponement of the determination 

 of the absolute displacement of the moving system until after its displacement relative 

 to a fixed line has been previously determined. The three problems which may be said 

 to form a natural (not merely a historically connected) group, and which offer the most 

 important illustrations of the class in question, are those of the rotation of a free body, 

 of the motion of a particle attracted to two fixed centres of force, and the problem of 

 three bodies. In the first and third of these, the invariable line is a line perpendicular 

 to the invariable plane, determinable by composition of the momenta of the several 

 elements of the system at any instant of time. In the second the invariable line is the 

 line joining the fixed centres ; and the distances of the moving point from the two fixed 

 centres or the angles which they make with the line of centres may be expressed by 

 equations complete within themselves, and into which the position of the plane con- 

 taining the moveable point and the fixed line does not enter. So again in the problem 

 of three bodies, without having recourse to the methods of defonnation employed by 

 Jacobi, and those who have followed in his track in treating the question, it is obrious, 

 a priori, that one integral may be gained, in the sense of one less being required, by 

 forming a system of equations from which the position of the intersection of the plane 

 of the three bodies with the invariable plane is excluded, equivalent in effect to the 

 so-called " elimination of the node " on Jacobi's method ; in which, however, the node so 



