299] DEVELOPMENTS OF THE DYNAMICAL THEORY. 319 



X 1 = 0, X 3 = 0, X s = 0. Combining these conditions with the last, we draw 

 the general conclusion that 



or 



&c. 



Thus we demonstrate that if a pair of screws 6, $&amp;gt; satisfy the six conditions, 

 they stand to each other in the relation of impulsive screws and instantaneous 

 screws. 



299. Cylindroid Reduced to a Plane. 



Suppose that the family of rigid bodies be found which make a, 77 and 

 /?, impulsive and instantaneous. Let there be any third screw, 7, and let 

 us seek for the locus of its impulsive screw, , for all the different rigid 

 bodies of the family. 



must satisfy the four equations 



cos (777) + / cos (a?) = 2sr ay , 



cos (arj) cos 



-^008(^0 = 

 COS (yC) 



cos (ar;) Y&amp;gt;) cos 



As there are four linear equations in the coordinates of , we have the 

 following theorem. 



If a, 77 and /3, f be given pairs of impulsive and instantaneous screws, 

 then the locus of the impulsive screw corresponding to 7, as an instan 

 taneous screw, is a cylindroid. 



But this cylindroid is of a special type. It is indeed a plane surface 

 rather than a cubic. The equations for can have this form : 



cos (a?) = A cos (yf), vr a( = C cos (y ), 



COS (f ) = B COS (y), CT^ = D COS (yf ), 



in which A, B, C, D are constants. 



The fact that cos (af) and cos (7^) have one fixed ratio, and cos (/3) and 

 cos (yf) another, shows that the direction of is fixed. The cylindroidal 

 locus of , therefore, degenerates to a system of parallel lines. 



At first it may seem surprising to find that CT of is constant. But the 



