THE PRINCIPAL SCREWS OF INERTIA. 47 



the moment of the given couple or 



/ 



whence i)\ = ^ i . 



aM 



The effect of an impulsive wrench on wi is, therefore, 

 to give the body a velocity of translation parallel to OA, 



and equal to -r- f w/', and also a velocity of rotation about 



OA equal to ^-, to/'. These movements unite to form a 

 aM 



twisting motion about a screw on OA, of which the pitch 

 is found by dividing the velocity of translation by the 

 velocity of rotation to be equal to a, this being the pitch 

 of wi, proves that an impulsive wrench on wi will make 

 the body commence to twist about wi, and that, therefore, 

 wi is a principal screw of inertia. Similar reasoning 

 applies to the remaining five screws. 



53. Impulsive Screws and Instantaneous Screws. If a 

 free quiescent rigid body receive an impulsive wrench 

 on a screw rj, the body will immediately commence to 

 twist about an instantaneous screw a. The co-ordinates 

 of a being given, it is required to determine the co- 

 ordinates of jj. 



The impulsive wrench on ?> of intensity r\" is to be 

 decomposed into components of intensities r\ f/ in, &c. 

 V'lje on wi, &c w 6 . The component on w will generate 

 a twist velocity about w amounting to 



but if 'a! be the twist velocity about a which is finally 

 produced, the expression just written must be equal to 

 ' a n , and hence we have the following useful result : 

 If the co-ordinates of the instantaneous screw be proper- 



