THE PRINCIPAL SCREWS OF INERTIA. 49 



Precisely the same condition must be satisfied when ]3 and 

 ij are reciprocal, and hence the general property of con- 

 jugate screws of inertia is true, whether the body be free 

 or constrained in any way. 



55. The Determination of the Impulsive Screw, corres- 

 ponding to a given instantaneous screw, is a definite 

 problem when the body is perfectly free. If, however, 

 the body be constrained, we shall show that any screw 

 selected from a certain screw complex will fulfil the re- 

 quired condition. 



Let B l9 &c. jB 6 _ n be 6 - n screws selected from the 

 screw complex which is reciprocal to that corresponding 

 to the freedom of the n th order possessed by the rigid 

 body. Let S be the screw about which the body is to 

 twist. Let X be any screw, an impulsive wrench about 

 which would make the body twist about S; then any 

 screw Y belonging to the screw-complex of the (7 - n} th 

 order, specified by the screws, X, B ly &c. -Z? 6 . n is an im- 

 pulsive screw, corresponding to S as an instantaneous 

 screw. For the wrench on Y may be resolved into 7-72 

 wrenches on X, B\ y &c. -# 6 -n ; of these, all but the first are 

 instantly destroyed by the reaction of the constraints, so 

 that the wrench on Kis practically equivalent to the 

 wrench on X, which, by hypothesis, will make the body 

 twist about S. 



For example, if the body had freedom of the fifth 

 order, then an impulsive wrench on any screw on a cer- 

 tain cylindroid will make the body commence to twist 

 about a given screw. 



If a body have freedom of the third order, then the 

 "locus" of an impulsive wrench which would make the 

 body twist about a given screw consists of all the screws 

 in space which are reciprocal to a certain cylindroid. 



56. System of Conjugate Screws of Inertia. We shall 

 now show that from the screw-complex of the n th order P, 



E 



