192] FREEDOM OF THE THIRD ORDER. 191 



191. Reaction of the Constraints. 



An impulsive wrench on a screw 77 acts upon a body with freedom of 

 the third order, and the body commences to move by twisting upon a screw 

 6. It is required to find the screw X, a wrench on which constitutes the 

 initial reaction of the constraints. Let &amp;lt;/&amp;gt; denote the impulsive screw which, 

 if the body were free, would correspond to 6 as the instantaneous screw. 

 Then \ must lie upon the cylindroid (&amp;lt;/&amp;gt;, 77), and may be determined by 

 choosing on (&amp;lt;, 77) a screw reciprocal to any screw of the given screw 

 system. 



192. Impulsive Screw is Indeterminate. 



Being given the instantaneous screw 6 in a system of the third order, 

 the corresponding impulsive screw 77 is indeterminate, because the impulsive 

 wrench may be compounded with any reactions of the constraints. In fact 

 77 may be any screw selected from a screw system of the fourth order, which 

 is thus found. Draw the diametral plane conjugate to a line parallel to 6 

 in the ellipsoid of inertia, and construct the cylindroid which consists of 

 screws belonging to the screw system parallel to this diametral plane. 

 Then any screw which is reciprocal to this cylindroid will be an impulsive 

 screw corresponding to 6 as an instantaneous screw. 



Thus we see that through any point in space a whole cone of screws can 

 be drawn, an impulsive wrench on any one of which would make the body 

 commence to twist about the same screw. 



One impulsive couple can always be found which would make the body 

 commence to twist about any given screw of the screw system. For a 

 couple in a plane perpendicular to the nodal line of a cylindroid may be 

 regarded as a wrench upon a screw reciprocal to the cylindroid ; and hence 

 a couple in a diametral plane of the ellipsoid of inertia, conjugate to the 

 diameter parallel to the screw 6, will make the body commence to twist 

 about the screw 6. 



It is somewhat remarkable that a force directed along the nodal line of 

 the cylindroid must make the body commence to twist about precisely the 

 same screw as the couple in a plane perpendicular to the nodal line. 



If a cylindroid be drawn through two of the principal screws of inertia, 

 then an impulsive wrench on any screw of this cylindroid will make the 

 body commence to twist about a screw on the same cylindroid. For the 

 impulsive wrench may be resolved into wrenches on the two principal 

 screws. Each of these will produce a twisting motion about the same 

 screw, which will, of course, compound into a twisting motion about a screw 

 on the same cylindroid. 



