210 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION, 



ft 



make the resolution at any other point, 0', the couple to be com- 

 pounded with S at 0', is perpendicular to E and 00' 7 so that if S 



has any component parallel to E it 

 cannot be neutralized by the new 

 couple. Accordingly in order that the 

 couple may vanish for any point O f , 

 the couple S must be perpendicular 

 to E at all other points. As a change 

 of introduces only a component 

 of S perpendicular to E, the com- 

 ponent parallel to E is unchanged. 

 Therefore the projection of S on E 

 is the same for all points 0, 



Fig. 56. 



Although in general E and S have different directions, we may 

 find points 0' for which they have the same direction. Let S and E 

 include the angle & at 0. Resolve S into S Q = S cos # 

 parallel to JR, and 8^ = 8 sin # perpendicular to .R. 

 If we take 0' on a line perpendicular to SE at a 

 distance ^ such that d - E = S sin # in the positive 

 direction of translation corresponding to a rotation 

 from E to $, the component 5 t will be neutralized, 

 and we shall have at 0' } E and S' = $ in the same 

 direction. This property holds for all points on the 

 line of E through 0'. This line is called Poinsot's 

 central axis. 



In order to consider the resolution at any point 

 we may refer it to the central axis. Drop a per- 

 pendicular from (Fig. 58) on the central axis, 

 and take this perpendicular for the axis of X, the 

 central axis for the axis of Z. 



Fig. 57. 



Then as above 



9) 

 and if xyz are the coordinates of the end of S, we have 



-i /^\ a -r) _ E 



10) z = /S , ?/ *ta> tan ^ = x - -> 



and for any point on the line of /S, 



11) = ^~-> or ## = -^ y, 

 y E x E yj 



that is the line of 8 lies on a hyperbolic paraboloid. 



