122-123] IMPULSIVE COUPLE. 181 



which is fixed in space, with an angular velocity proportional to the length 01 

 of the radius vector drawn from the origin to the point of contact /. The 

 representation of the actual motion is then completed by impressing on the 

 whole system of rolling ellipsoid and plane a velocity of translation whose 

 components are given by (x). This velocity is in the direction of the 

 normal OM to the tangent plane of the quadric 



at the point P where 01 meets it, and is equal to 

 e 3 



OP ojf xangular velocit y f 



When 01 does not meet the quadric (xiii), but the conjugate quadric obtained 

 by changing the sign of e, the sense of the velocity (xiv) is reversed*. 



123. The problem of the integration of the equations of 

 motion of a solid in the general case has engaged the attention of 

 several mathematicians, but, as might be anticipated from the 

 complexity of the question, the meaning of the results is not 

 easily grasped. 



In what follows we shall in the first place inquire what 

 simplifications occur in the formula for the kinetic energy, for 

 special classes of solids, and then proceed to investigate one or 

 two particular problems of considerable interest which can be 

 treated without difficult mathematics. 



1. If the solid has a plane of symmetry, as regards both its 

 form and the distribution of matter in its interior, then, taking this 

 plane as that of xy, it is evident that the energy of the motion is 

 unaltered if we reverse the signs of w, p, q, the motion being 

 exactly similar in the two cases. This requires that A', B', P', Q', 

 L, M, L f , M', N" should vanish. One of the directions of perma- 

 nent translation is then parallel to z. The three screws of Art. 122 

 are now pure rotations ; the axis of one of them is parallel to z ; 

 the axes of the other two are at right angles in the plane xy, but 

 do not in general intersect the first. 



2. If the body have a second plane of symmetry, at right 

 angles to the former one, let this be taken as the plane of zx. 

 We find, in the same way, that in this case the coefficients 



* The substance of this Art. is taken from a paper, " On the Free Motion of a 

 Solid through an Infinite Mass of Liquid," Proc. Lond. Math. Soc. t t. viii. (1877). 

 Similar results were obtained independently by Craig, " The Motion of a Solid in a 

 Fluid," Amer. Journ. of Math., t. ii. (1879). 



