292 PROFESSOR TAIT ON THE ROTATION OF A 



moving body, the vector P , which is always in the plane through the fixed point 

 and perpendicular to 7, belongs to a cone of which 7 is a normal, and which there- 

 fore rolls on that plane. But the cone also slides, because the vector p which is 

 in contact with the plane is not the instantaneous axis of the body. This con- 

 struction for the illustration of the motion is also due to Poinsot, and the com- 

 plete analytical solution of the problem has been given, from this point of view, 

 by Rueb and Jacobi * It is easy to see that the angular velocity of the sliding 

 motion is the constant resolved angular velocity of the body about the fixed line 7, 

 which has the value 



45. When two of the moments of inertia of the rigid body are equal, i. e when 

 the symbolical cubic in <p or 9 has two equal roots, all the previous dynamical 

 work becomes immensely simplified. In fact, if we now take a, /3, 7 as unit- 

 vectors coinciding with the principal axes of the moving body, we have by (23) 



9 g = - AaSetg - B/SSjSg - BySyg . 



But 



s = - aSag - /3S/% - 7 S 7S , 



so that 



Pg = Bg-(A-B)«S«g ..... (35^ 



and thus depends upon the position of the one vector a. We may attempt to 

 determine the motion without at first introducing the consideration of th 

 quaternion which has been our principal object of study in this paper. 



46. The general equation of § 24 



becomes, by substituting for <p from (35), 



Bi-.(A~B)«S«=-(A-B)VasS« . . . (36 ) 



e 



Operating by S . a , we have 



37). 



Omitting, therefore, this term from (36) and operating by S . £ , we have 



s« = , 



whose integral is 



2 



2- — constant = — Q 2 , suppose, ... (-33) 



But we have always by § 1 



because a is fixed in the body. 

 From this we see that 



a = V 



Sea = . 

 * See Cayley, B. A. Report, 1862. 



