257] HOMOGRAPHIC SCREW SYSTEMS. 271 



256. Correspondence of m and n systems . 



We may look at the matter in a more general manner. Consider an 

 w-system (J.) of screws, and an w-system (B) (m&amp;gt;n). (If we make ra = 6 

 and n = 2, this system includes the system we have been just discussing.) 

 To one screw in A will correspond one screw in B, but to one screw in B 

 will correspond, not a single screw in A, but an (m + 1 ?i)-system of screws. 



If m = n, we find that one screw of one system corresponds to one screw 

 of the other system. Thus, if m = n = 2, we have a pair of cylindroids, and 

 one screw on one cylindroid corresponds to one screw on the other. If 

 m = 3, and n = 2, we see that to each screw on the cylindroid will cor 

 respond a whole cylindroid of screws belonging to the three-system. For 

 example, if a body have freedom of the second order and a screw be indicated 

 on the cylindroid which defines the freedom, then a whole cylindroid full of 

 screws can always be chosen from any three-system, an impulsive wrench on 

 any one of which will make the body commence to twist about the indicated 

 screw. 



257. Screws common to the two systems. 



The property of the screws common to the two homographic systems 

 will of course require some modification when we are only considering an 

 wi-system and an ?i-system. Let us take the case of a three-system on the 

 one hand, and a six-system, or all the screws in space, on the other hand. 

 To each screw a of the three-system A must correspond, a four-system, B, 

 so that a cone of the screws of this four-system can be drawn through every 

 point in space. It is interesting to note that one screw /3 can be found, 

 which, besides belonging to B, belongs also to A. Take any two screws 

 reciprocal to B, arid any three screws reciprocal to A, then the single screw 

 /3, which is reciprocal to the five screws thus found, belongs to both A and 

 B. We thus see that to each screw a of A, one corresponding screw in the 

 same system can be determined. The result just arrived at can be similarly 

 shown generally, and thus we find that when every screw in space cor 

 responds to a screw of an ?i-system, then each screw of the n-system will 

 correspond to a (7 ?i)-system, and among the screws of this system one 

 can always be found which lies on the original n-system. 



As a mechanical illustration of this result we may refer to the theorem 

 ( 96), that if a rigid body has freedom of the nth order, then, no matter 

 what be the system of forces which act upon it, we may in general combine 

 the resultant wrench with certain reactions of the constraints, so as to 

 produce a wrench on a screw of the n-system which defines the freedom of 

 the body, and this wrench will be dynamically equivalent to the given 

 system of forces. 



