130 ON THE MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. V. 



These are identical in form with the equations of motion of a rigid 

 body about a fixed point, so that we may make use of Poinsot s 

 well-known solution of the latter problem. The angular motion 

 of the body is therefore obtained by making the ellipsoid (30), 

 which is fixed in the body, roll on the plane 



\x + py -f vz const., 



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 whose components are given by (32). 

 The direction of this velocity is that of the normal OM to the 

 tangent plane to the quadric 



at the point P where 01 meets this quadric, and its magnitude is 

 77T&amp;gt; x angular velocity of body ......... (35). 



If 01 do not meet (34), but the conjugate quadric obtained by 

 changing the sign of e, the sense of the velocity (35) is reversed. 



116. Of course for particular varieties of the moving solid the 

 expression for 2T becomes greatly simplified. For instance: 



(a) let us suppose that the body has a plane of symmetry 

 as regards both its form and the distribution of matter in its in 

 terior, and let this plane be taken as that of xy. It is plain 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 re 

 quires that A , B , P, Q , L, M, L t M , N&quot; should vanish. One 

 of the directions of permanent translation is then parallel to z. 

 The three screws of Art. 114 are now pure rotations; the axis of 

 one of them is parallel to z ; those of the other two are at right 

 angles in the plane ay, but do not in general intersect the first. 



(b) If the body have a second plane of symmetry, at right 

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

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



