Professor Mac Cullagh's Lectures on Rotation. 147 



VII. — The Axis of principal Moments isjixed in Space. 



This is evident from D'Alembert's principle, but may be shown by geome- 

 trical considerations in the particular case under consideration. The axis varies 

 in position in the body, in consequence of the centrifugal couple, which must be 

 compounded with the impressed couple at each instant. Referring to equa- 

 tion (8j, the value of the centrifugal couple is - fxw-PQdt, the principal mo- 

 ment being G = ixwPIi (vid. 9). Hence the angle through which the axis 



of principal moment shifts in an element of time is - -p~ ; this angle, mul- 

 tiplied by the constant radius vector, will give the elementary motion on 

 the spherical conic traced by the axis of principal moment on the surface of the 

 ellipsoid ; this motion is therefore - wQdt ; but in the same time the point 

 of the body which coincides with the point where the axis of moments pierces 

 the spherical conic will describe the angle + wQdt in consequence of the an- 

 gular rotation. Hence the axis of moments will remain fixed in space, and will 

 move in the body witli a velocity proportional to the tangent of the an" le be- 

 tween the radius vector and perpendicular, the motion beino- in a direction 

 opposite to the direction of the rotation ; this is evident from the consideration 



that Qu) = Pto tan <p, Pw being constant and equal to —jr (vid. 9). 



VIII. — To find the Motion of the principal Axis in the BodAj. 



First Method. 

 The point of the principal axis of moments, which is situated at the distance i? 

 from the centre, moves on the spherical conic \yhich has been determined. Let 

 this point be projected on the thi-ee co-ordinate planes ; then, since the spherical 

 conic is projected into a conic section, the movement of the axis of moments is 

 reduced to the movement of a point on a conic section, according to a law 

 which must be determined. The radius vector describes an elementary triangle 

 in the surface of the cone (11) ; let the projections of this triangle on the co- 

 ordinate planes be (dAi, dAn, dA,) ; we obtain easily 



u2 



