48 THE PRINCIPAL SCREWS OF INERTIA. 



tional to ai, &c. a 6 , then the co-ordinates of the correspond- 

 ing impulsive screw are proportional to p l ai, &c. / 6 a 6 . 



54. Conjugate Screws of Inertia. If a and j3 be two 

 instantaneous screws, and if r\ and be the correspond- 

 ing impulsive screws, then when a is reciprocal to % we 

 must have /3 reciprocal to rj. We shall first suppose the 

 body to be perfectly free. 



The co-ordinates of S are proportional to/i/3i, &c./ 6 j3 6 , 

 hence the condition that a and are reciprocal ( 34) is 



/i 2 ai |3i + &c. + / 6 2 a 6 /3 6 = o. 



But this is precisely the equation which we should 

 have found by expressing the condition that ]3 and rj were 

 reciprocal. 



When this relation is fulfilled, the screws a and ]3 are 

 said to be conjugate screws of inertia. 



We shall now show that this theorem will still remain 

 true even if the body be only partially free. When the 

 body receives an impulsive wrench on there is an im- 

 pulsive reaction of the constraints on a screw ju . The 

 effect on the body is, therefore, the same as if it had been 

 free, but had received an impulsive wrench of which the 

 component on o>i had the intensity /x i + ju"/^ ; hence, 

 h being a constant, we have 



&c. &c. 



", + juV 



multiplying the first of these equations by p l a\, the 

 second by/ 2 2, &c., adding the six products, and re- 

 membering that a and are reciprocal by hypothesis, 

 while a and /it are reciprocal, by the nature of the re- 

 actions of the constraints ( 43), we have, as before 



