190] FREEDOM OF THE THIRD ORDER. 189 



188. Lemma. 



If from a screw system of the nih order we select n screws A ly ... , A n , 

 which are conjugate screws of inertia ( 87), and if 8 l be any screw which 

 is reciprocal to A 2 ,...,A n , then an impulsive wrench on S Y will cause the 

 body, when only free to twist about the screws of the system, to commence 

 to twist about A lt Let 7^ be the screw which, if the body were perfectly 

 free, would be the impulsive screw corresponding to A l as the instantaneous 

 screw. R! must be reciprocal to A.,,...,A n ( 81). Take also 6 n screws 

 of the reciprocal system B 1 ,...,B ti ^ n . Then the 8 n screws R l} S lt B l} ... , 

 B 6 _ n must be reciprocal to the n 1 screws A 2 , ...,A n , and therefore the 

 8 n screws must belong to a screw system of the (7 ?i)th order. Hence 

 an impulsive wrench upon the screw Si can be resolved into components on 

 RI, BI, ... ,B 6 - n . Of these all but the first are neutralised by the reactions 

 of the constraints, and by hypothesis the effect of an impulsive wrench 

 upon Q is to make the body commence to twist about A 1} and therefore 

 an impulsive wrench on Si would make the body twist about AI. 



189. Relation between the Impulsive Screw and the Instan 

 taneous Screw. 



A quiescent rigid body which possesses freedom of the third order is 

 acted upon by an impulsive wrench about a given screw 77. It is required 

 to determine the instantaneous screw 6, about which the body will commence 

 to twist. 



The screws which belong to the system, and are at the same time reci 

 procal to 77, must all lie upon a cylindroid, as they each fulfil the condition 

 of being reciprocal to four screws. All the screws on the cylindroid are 

 parallel to a certain plane drawn through the centre of the pitch quadric, 

 which may be termed the reciprocal plane with respect to the screw 77. The 

 reciprocal plane having been found, the diameter conjugate to this plane 

 in the ellipsoid of inertia is parallel to the required screw 6. 



For let fi and v denote two screws of the system parallel to a pair of 

 conjugate diameters of the ellipsoid of inertia in the reciprocal plane. Then 

 6, fi, v are a triad of conjugate screws of inertia ; but 77 is reciprocal to //, 

 and v, and, therefore, by the lemma of the last article, an impulsive wrench 

 upon i] will make the body commence to twist about 6. 



190. Kinetic Energy acquired by an Impulse. 



We shall now consider the following problem : A quiescent rigid body 

 of mass M receives an impulsive wrench of intensity rf&quot; on a screw 77. We 

 have now to determine the locus of a screw belonging to a screw system 

 of the third order, such that, if the body be constrained to twist about 6, it 



