85] 



POLHODE AND HERPOLHODE. 



263 



This is the equation of the cone passing through the polhode, with 

 its vertex at the fixed point, that is the rolling or polhode cone. 



We find then that the rolling cone for a body moving under 

 no forces is of the second order. If it is to be real, we must have 



41) A^^C, 



that is the perpendicular must have a length intermediate between 

 the greatest and least axes of the ellipsoid. If -^ = A the cone is 



42) 



representing a pair of imaginary planes, intersecting in the real line 

 y = 3 = 0, the X-axis. Thus in this case the rolling cone reduces 



to a line, fixed both in the body and in space. If ^ = C, we have 



a similar result. If 



we 



43) 



A(A- B)x* - 



representing two real planes intersecting 

 in the Y-axis, and making an angle with 

 the XY- plane whose tangent is 



44) 



__ 

 x ~ - 



C(S-C) 



These are the planes which separate the 

 polhodes surrounding the end of the major 

 axis from those about the minor axis. The 

 polhodes are twisted curves of the fourth 

 order, whose appearance is shown in 

 perspective in Fig. 77. The separating 

 polhodes are drawn black. 



Since the polhode is a closed curve, 

 the radius vector of a point on it oscillates 

 between a maximum and a minimum 

 value. If 6 is the distance of a point 

 on the herpolhode from the foot of the 



perpendicular d, since 6 2 = g 2 d 2 , 6 oscillates between two constant 

 values, and the herpolhode is tangent to two circles. Since the 

 polhode is described periodically the various arcs of the herpolhode 

 corresponding to repetitions of the polhode are all alike. The her- 

 polhode is not in general a reentrant curve. The name herpolhode 

 was given by Poinsot from the verb SQXSW, to creep (like a snake) 

 from the supposedly undulating nature of the curve, it has however 



Fig. 77. 



