262 VII. DYNAMICS OF ROTATING BODIES. 



Multiplying now by Ap } Bq, Cr, and adding, 



34) A * p % + B . q <M + C 'r-0, 



which is integrated as 



35) A*p* + -# V + C V = const. 



But this, since the left-hand member is equal to i? 2 , is the equation 

 of conservation of angular momentum. The equation alone does not 

 show the fixity of the direction of H in space. 



The point P in which the instantaneous axis intersects the 

 ellipsoid of inertia at the fixed point is called the pole of the 

 instantaneous axis. Its coordinates are 



op oq or 



x = ^-, y = > z = ~- 



CO 00 0) 



Now the length of the perpendicular d is, since it is the projection 

 of Q on the direction of the normal, 



36) d = x cos (nx) -f y cos (ny) -f z cos (nz) 



snce 



2T -*- 



Accordingly since T and H are constant, d is constant, and the 

 tangent plane being perpendicular to the invariable line H is fixed 

 in space. Poinsot called the locus of the pole of the instantaneous 

 axis on the ellipsoid, the polhode (rt6ho$ axis, bd6$ path), and its 

 locus on the tangent plane the herpolhode. 

 The ellipsoid of inertia being 



37) Ax 2 + W + Cz* = 1, 



the distance of the tangent plane at x, y, z from the center is 



Since this is to be constant, this equation with that of the ellipsoid 

 define the polhode curve. Combining the equation 



39) A*x* + &f + C*s* = -~ 



with that of the ellipsoid, divided by d 2 , we obtain by subtraction 



40) A -Ax* + B -Bf+ C - CS = 0. 



