ACTED ON BY NO EXTERNAL FOECES. 771 



contrafocal, be set in a pai-allel position at rest, and are acted on by two equal and 

 coaxial but contrary impulsive couples, their principal axes will continue throughout 

 the motion to make equal but contrary angles with the invariable line, and will admit 

 of being brought back to a position of parallelism by means of a rotatory displacement 

 about the invariable line proportional to the time. Thus, leaving out of consideration 

 this displacement, correlated solid bodies (as those ihay be termed whose kinematical 

 exponents are confocal ellipsoids) may be made to move equally and similarly, and 

 contrarelated ones (as we may term those whose kinematical exponents are contrafocal 

 ellipsoids) equally and contrarily without the action of any external force. It will 

 eventually be seen that there is a practical advantage in considering L as retaining the 

 same sign in both cases, and throwing the contrariety of motion in the second case 

 upon the change of the inclinations a, |3, y into their supplements. 



Thus the motion of a body is arithmetically given when that of any other of the 

 series of those to whose kinematical exponents its own is either confocal or contrafocal 

 has been determined. 



Alike for the two cases of con- and contra-focalism it will be convenient to disrcarard 

 this uniform motion of rotation, treating it in the light merely of a coiTection*, so that 

 the motions of all the bodies contained in either one series may be considered in regard 

 to themselves as coincident, and as siqyplemental (in a sense that explains itself) in regard 

 of the motions of the bodies belonging to the other series. I shall now show as a corol- 

 lary from the above proposition that, with the above understanding, the motion of any 

 rigid body may (subject to an unimportant exception that will be stated in its proper 

 place) be made identical with that of one real indefinitely flattened disk, and supplemental 

 to that of another. The case of a disk, it will be noticed, is that in which one of the 

 principal moments of inertia becomes equal to the sum of the other two ; in general 

 these moments of inertia must not only be positive, but each must be not greater than 

 the sum of the other two, as is the case with the lengths of the sides of a triangle ; in 

 the extreme case, when the body is reduced to but two dimensions, the greatest becomes 

 equal to the sum of the other two, and conversely, when this is so, the body can only 

 be of the form of a flat disk ; the above is ob\ious when it is remembered that the 

 moments of inertia are the sums of the three intrinsically positive quantities 2/H.i-*, 

 2??iy, Stjiz'^ taken two and two together. So also it is well to notice that the modular 



quantity p m equation (2) is not absolutely arbitrary, but besides being essentially 

 positive, is conditioned to lie between the least and greatest of the quantities \; i ; 1, 



since otherwise the quantities (cos a)' ; (cos fif ; (cos yf in equations (1) and (2) could 

 not all remain positive, and consequently such equations would not correspond to any 

 real case of motion. 



• The apparent motions of any two correlated or contrarelated bodies to two spectators standing respectively 

 on the invariable plane of each may be made identical or similar, provided a certain uniform angular velocity 

 be imparted to one of these planes. 



