72 THE THEORY OF SCREWS. [81 



the six products and remembering that a and are reciprocal by hypothesis 

 while a. and /* are reciprocal by the nature of the reactions of the constraints, 

 we have 



The symmetry of this equation shows that in this case 77 must be reciprocal 

 to ft. Hence we have the following theorem which is of fundamental import 

 ance in the subject of the present volume. 



If a. be the instantaneous screw about which a quiescent rigid body either 

 perfectly free or constrained in any manner whatever commences to twist in 

 consequence of an impulsive wrench on some screw tj, and if ft be another 

 instantaneous screw, similarly related to an impulsive screw , then whenever % 

 is reciprocal to a we shall find that 77 is reciprocal to ft. 



When this relation is fulfilled the screws a and ft are said to be conjugate 

 screws of inertia. 



82. The Determination of the Impulsive Screw, corresponding 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 system will, in general, fulfil the required 

 condition. 



Let B lf ... B n _ n be 6 n screws selected from the screw system which is 

 reciprocal to that corresponding to the freedom of the nth order possessed by 

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

 X be any one of the screws, an impulsive wrench about which would make 

 the body twist about S ; then any screw Y belonging to the screw system of 

 the (7 - n)th order, specified by the screws, X, B 1} ... B 6 _ n is an impulsive 

 screw, corresponding to S as an instantaneous screw. For the wrench on Y 

 may be resolved into 7 n wrenches on X, B 1} ... B 6 _ n ; of these, all but 

 the first are instantly destroyed by the reaction of the constraints, so that the 

 wrench on Y is practically equivalent to the wrench on X, which, by hypo 

 thesis, will make the body twist about S. 



As an example : if the body had freedom of the fifth order, then an 

 impulsive wrench on any screw on a certain cylindroid will make the body 

 commence to twist about a given screw. 



As another example : if a body have freedom of the third order, then 

 the &quot;locus&quot; 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. 



83. System of Conjugate Screws of Inertia. 



We shall now show that from the screw system of the nth order P, which 

 expresses the freedom of the rigid body, generally n screws can be selected 



