774 PROFESSOR SYLVESTER ON THE MOTION OF A RIGID BODY 



or 



(2A-C>=AC, ^=2^. 



and 



__A/A AC _ AC . 



■^'-fi-A— 2{C-A)' *^'— C,-A' 



so that the reduction will be proper or improper according as the unequal moment 

 of inertia is greater or less than either of the equal ones. 



A relation has been obtained geometrically in the commencement of this memoir 

 between the squared velocities of any two dynamically equivalent bodies represented by 

 confocal ellipsoids. To complete the theory, it is proper to find the exact nature of this 

 relation when a given body has been reduced to a disk, whether by the direct or supple- 

 mental method. 



First, in the case of direct reduction, using u,, Wj, Uj for the angular velocities of the 

 disk, and «y,, <a^, U3 for those of the associated body in corresponding positions about the 

 principal axes, and v, u for the total angular velocities of the disk and body respecetively, 



L L _ L 



u, = T~cosa, U3=jT-cosp, Ug^^cosy, 



L L ^ L 



<y,=xCosa, oi).^-=^cos,p, 6).^=-^^ COS y, 



Aj— A~^' Bi— B ^' C,— G ^• 



Hence 



»=2a<?=2('f^+lA) (cos a)^=Si^f +2PX2 ^^^+PX' 



= y^ + 2DX^^2 + L=X^ 



or 



..■=XL-(f+x). 



And again, in the case of supplemental reduction, using y, , y^ , 1*3, v for the partial and 

 total angular velocities of the disk, 



L L - L 



Ui = — v-cosa, Uj= — ^cosp, ^3= — ^cosy, 



1 1 \ \ \ _ I 



M — ^—K' B'— ''~B' C^— ''~C' 



M 



a;''=2('lA-^') (cosa)' = y^-2PXp+LV, 



or 



2M 



a,=-y^=XL^(-^+x) ; 



showing that in both cases alike the differences between the squared velocity of the 

 body and that of either its representative disks is constant throughout the motion, as 

 might also have been predicted a ^n'on from the form of the elliptic function connecting 



