136] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 



but, from the property of the circle, 



B Y. XO = AX . BO ; A Y . XO = EX . A ; 

 whence 



AB .XY. XO =AX\ AB. BO + BX&amp;gt;.AB.AO , 



from which we obtain, as before, 



125 



136. Impulsive Screws and Instantaneous Screws. 



A rigid body having two degrees of freedom lies initially at rest. It is 

 suddenly acted upon by an impulsive wrench of large intensity acting for a 

 short time. The body will, in general, commence to move by twisting about 

 some screw on the cylindroid, and the kinetic problem now to be studied is 

 the following : Given the impulsive screw, and the intensity of the impul 

 sive wrench, find the instantaneous screw and the acquired twist velocity. 



The problem will be rendered more concise by the conception of the 

 reduced wrench ( 96). It is to be remembered, that as the body is only 

 partially free, there are an infinite number of screws on which wrenches 

 would make the body commence to twist about a given screw on the cylin 

 droid. For, let 6 be an impulsive screw situated anywhere, and let an 

 impulsive wrench on cause the body to commence to move by twisting 

 about some screw, a, on the cylindroid. Let \, /u-, v, p be any four screws 

 reciprocal to the cylindroid. Then any wrench on a screw belonging to the 

 system defined by these five screws will make the body commence to move 

 by twisting about a. Let e be that one screw on the cylindroid which is 

 reciprocal to 6, then e is reciprocal to the whole system defined by A,, /*, v, p, 

 0, and, conversely, each screw of this system will be reciprocal to e. We 

 thus see that any screw, wherever situated, provided only that it is 

 reciprocal to e, will be an impulsive screw corresponding to a as an instan 

 taneous screw. Any one of this system may, with perfect generality, be 

 chosen as the impulsive screw. Among them there is one which has a 

 special feature. It is that screw, &amp;lt;/&amp;gt;, on the cylindroid which is reciprocal to 

 77 ; and hence we have the following theorem ( 128): 



Given any screw, a, on the cylindroid, then there is in general another screw, 

 &amp;lt;/&amp;gt;, also on the cylindroid, such that an impulsive wrench administered on &amp;lt;f) 

 will make the body twist about a. 



This correspondence of the two systems of screws must be of the one-to- 

 one type ; for, suppose that two impulsive screws on the cylindroid had the 

 same instantaneous screw, it would then be possible for two impulsive 

 wrenches, of properly chosen intensities on two different screws, to produce 



