of a rigid Body round affixed Point, 455 



these angles their values given by (1.) and (2.), we find 

 K G cos V = (w 3 — co 3 ) .fy zdm .fx z d m = 0, 



hence V = 90, or the plane of G passes through the axis of K. 



VI. To express G in terms of K, a>, and the moment of 

 inertia round the instantaneous axis of rotation. 



Squaring the six equations (1.) (2.), adding the first three 

 together, and also the last three, we find 



K 2 = <a*{fxz dmf + u? [fy z dm)* + w 2 (/Cr 2 +,y 2 ) dm)* 



G 2 = «» 4 (fx z dm)* + u> 4 (fy z dm) 9 . 



Multiplying the first of these equations by up, subtracting 

 the second from the first, and calling the moment of inertia 

 round the instantaneous axis of rotation D, we get 



G 2 m K 2 c« 2 -co 4 D 2 (3.) 



VII. Let «/ denote the angular velocity round any line 

 which makes the angle 9 with the instantaneous axis of rota- 

 tion, oo the angular velocity round this axis, then «/ = w cos 6. 



o 



Let O A be the instantaneous axis, O B a line passing 

 through the fixed point O, making the angle 6 with the former 

 O C, C A right lines perpendicular to OB, O A ; let the 

 point C move to c, a point indefinitely near the former, then 

 Cc= OC.«/, andCc = C A. «; hence O C. «/ =CA.», 

 but C A = O C cos 6; hence of = oo cos 6. 



VIII. Conceive an ellipsoid whose centre coincides with 

 the origin, whose axes coincide with the principal axes of 

 rotation of the body passing through the fixed point, and the 

 squares of whose semiaxes are proportional to the moments 

 of inertia round these axes, so that 



A = ri A a\ B = n 3 b\ C = w 3 c* . . . (4.) 



n being a constant, a, b, c the semiaxes of the ellipsoid, and 

 ABC the moments of inertia round those axes. 



2F2 



