of a rigid Body round a fixed Point. 437 



Now the cosines of the angles which the perpendicular to the 

 tangent plane touching the surface at the point {xyz) makes 



Vx Py ~Pz 



with the axes of coordinates, are also — S - J -r#, -o-; hence 



a 1 o l c l 



the axis of instantaneous rotation coincides with this perpen- 

 dicular. 



XI. During the whole motion of rotation the semidiameler 

 of the ellipsoid perpendicular to the plane of the impressed mo- 

 ment is invariable f or u is constant; to show this we shall 

 make use of the following property of the ellipsoid : — 



Let a tangent plane be drawn to an ellipsoid, u the semi- 

 diameter through the point of contact, P the perpendicular 

 on this tangent plane from the centre, m the semidiameter 

 of the ellipsoid perpendicular to the plane of u and P; m 

 and u are the semiaxes of the section of the surface made by 

 the plane containing u and m. 



As P is perpendicular to the tangent plane, every plane 

 which passes through this line is perpendicular to this plane, 

 hence the plane containing u and P is perpendicular to the 

 tangent plane. 



In like manner, as the semidiameter m of the ellipsoid is 

 perpendicular to the plane of u and P, the plane which passes 

 through m and u t is perpendicular to the plane containing 

 U and P ; hence the tangent plane, and the plane of m and 

 «, are each perpendicular to the plane of u and P, therefore 

 their intersection is perpendicular to the same plane, and 

 therefore parallel to m, and therefore perpendicular to u\ but 

 when a tangent to a conic section is perpendicular to the 

 diameter passing through the point of contact, this diameter 

 is an axe of the section; therefore wand m are the semiaxes of 

 the section of the ellipsoid containing,?* and m. 



Now u being the axis of the impressed moment, and P the 

 perpendicular on the tangent plane, through the vertex of u, 

 P is the axis of instantaneous rotation, therefore the plane of 

 u and P is the plane of the centrifugal moment, and m per- 

 pendicular to this plane (V) is its axis. 



Assume a point s, on this line m, so that o s may be to u 

 as G to K ; and complete the rectangle OsQv, the diagonal 

 o v of this rectangle will represent both in magnitude and di- 

 rection, the axis of the resultant moment at the end of the 

 first instant ; during this instant the vertex of the axe of the 

 resultant moment has travelled on the surface of the ellip- 

 soid, and also on the surface of a concentric sphere, whose 

 radius is w, since the line Q v perpendicular to O Q or u is 

 a tangent both to this sphere and the ellipsoid ; hence at the 



