314] THE GEOMETRICAL THEORY. 339 



But it is of the essence of the theory that the reactions of the constraints 

 by which the motion of the body is limited to twists about screws on the 

 cylindroid A must be wrenches on the reciprocal system B. So far, therefore, 

 as the body thus constrained is concerned, the reactions of the constraints 

 will neutralize the wrench on TJ I . Thus the wrench on r} 2 is the only part of 

 the impulsive wrench which need be considered. 



But we already know from the construction that an impulsive wrench on^ 

 will produce an instantaneous twist velocity about a determined screw a on 

 A. Thus we have found the initial movement, and the investigation is geo 

 metrically complete. 



313. Freedom of the Third Order. 



We next consider the case in which a rigid body has freedom of the third 

 order. We require, as before, to find a geometrical construction for the in 

 stantaneous screw a corresponding to a given impulsive screw 77. 



Let A be the system of screws of the third order about which the body 

 is free to twist. Let B be the system of screws of the third order reciprocal 

 to A. We must first construct the system of the third order P which 

 consists of the impulsive screws that would have made the body, if perfectly 

 free, twist about the several screws of A. 



As already explained ( 307) we can, in one way, but only in one way, 

 resolve the original wrench on 77 into wrenches 7^ on B, and 772 on P. The 

 former is destroyed by the reactions of the constants. The latter gives rise 

 to a twist velocity about a determinate screw on A. Thus the problem has 

 been solved. 



314. General Case. 



We can obviously extend a similar line of reasoning to the cases where 

 the body had freedom of the remaining degrees. It will, however, be as simple 

 to write the general case at once. 



Let A be a system of screws of the ?ith order, about which a body is free 

 to twist, any other movements being prevented by constraints. If the body 

 receive an impulsive wrench, on any screw 77, it is required to determine the 

 instantaneous screw, of course belonging to A, about which the body will 

 commence to twist. 



Let B be the system of screws of the (6 - w)th order, reciprocal to A. 

 The wrenches arising from the reaction of the constraints must, of course, be 

 situated on the screws of the system B. 



Let P be the system of screws of the nth order, which, in case the body 



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