95j THE PRINCIPAL SCREWS OF INERTIA. 83 



93. Lemma. 



If a group of instantaneous screws belong to a system of the rath order, 

 then the body being quite free the corresponding group of impulsive screws 

 also belong to a system of the nth order ; for, suppose that n + 1 twisting 

 motions about n + 1 screws neutralise, then the corresponding n + 1 im 

 pulsive wrenches must equilibrate, but this would not be possible unless all the 

 impulsive screws belonged to a screw system of the ?ith order. 



94. Euler s Theorem. 



If a free or constrained rigid body is acted upon by an impulsive wrench, the 

 body will commence to move with a larger kinetic energy when it is permitted 

 to select its own instantaneous screw from the screw system P defining the 

 freedom, than it would have acquired, had it been arbitrarily restricted to 

 any other screw of the system. 



Let Q be the reciprocal system of the (6 w)th order, and let P be the 

 screw system of the nth order, consisting of those impulsive screws which, 

 if the body were free, would correspond to the screws of P as instantaneous 

 screws. 



Let i) be any screw on which the body receives an impulsive wrench. De 

 compose this wrench into components on a system of six screws consisting of any 

 n screws from P , and any 6 n screws from Q. The latter are neutralised by 

 the reactions of the constraints, and may be omitted, while the former com 

 pound into one wrench on a screw belonging to P ; we may therefore 

 replace the given wrench by a wrench on , If the body were perfectly free, 

 an impulsive wrench on must make the body twist about some screw a on 

 P. In the present case, although the body is not perfectly free, yet it is free 

 so far as twisting about a is concerned, and we may therefore, with reference 

 to this particular impulse about &quot;, consider the body as being perfectly free. 

 It follows from 92 that there would be a loss of energy if the body were 

 compelled to twist about any screw other than a, which is the one it naturally 

 chooses. 



95. Co-ordinates in a Screw System. 



The co-ordinates of a screw belonging to a specified screw-system can be 

 greatly simplified by taking n co-reciprocal screws belonging to the given 

 screw system as a portion of the six screws of reference. The remaining 

 6 n screws of reference must then belong to the reciprocal screw system. 

 It follows that out of the six co-ordinates a l} ... a tt of a screw a, which belongs 

 to the system, 6 n are actually zero. Thus we are enabled to give the more 

 general definition of screw co-ordinates which is now enunciated. 



If a wrench, of which the intensity is one unit on a screw a, which belongs to 



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