84 THE THEORY OF SCREWS. [95- 



a certain screw system of the nth order, be decomposed into n wrenches of 

 intensities a 1} ... a n on n co-reciprocal screws belonging to the same screw system, 

 then the n quantities a 1} . . . ^ are said to be the co-ordinates of the screw a. Thus 

 the pitch of a will be represented by p^ + . . . + p n a n *. The virtual coefficient 

 of a and /3 will be (p&fa + ... +p n &amp;lt;*n@n)- 



We may here remark that in general one screw can be found upon a screw 

 system of the wth order reciprocal to n 1 given screws of the same system. 

 For, take 6 n screws of the reciprocal screw system, then the required screw 

 is reciprocal to 6 n + n 1 = 5 known screws, and is therefore determined 

 ( 25). 



96. The Reduced Wrench. 



A wrench which acts upon a constrained rigid body may in general be 

 replaced by a wrench on a screiu belonging to the screw system, which defines 

 the freedom of the body. 



Take n screws from the screw system of the ?ith order which defines the 

 freedom, and 6 n screws from the reciprocal system. Decompose the given 

 wrench into components on these six screws. The component wrenches on 

 the reciprocal system are neutralized by the reactions of the constraints, and 

 may be discarded, while the remainder must compound into a wrench on the 

 given screw system. 



Whenever a given external wrench is replaced by an equivalent wrench 

 upon a screw of the system which defines the freedom of the body, the latter 

 may be termed, for convenience, the reduced wrench. 



It will be observed, that although the reduced wrench can be determined 

 from the given wrench, that the converse problem is indeterminate (n &amp;lt; 6). 



We may state this result in a somewhat different manner. A given 

 wrench can in general be resolved into two wrenches one on a screw of any 

 given system, and the other on a screw of the reciprocal screw system. The 

 former of these is what we denote by the reduced wrench. 



This theorem of the reduced wrench ceases to be true in the case when 

 the screw system and the reciprocal screw system have one screw in common. 

 As such a screw must be reciprocal to both systems it follows that all the 

 screws of both systems must be comprised in a single five-system. This is 

 obviously a very special case, but whenever the condition indicated is satisfied 

 it will not be possible to resolve an impulsive wrench into components on the 

 two reciprocal systems, unless it should also happen that the impulsive 

 wrench itself belongs to the five-system*. 



* I am indebted to Mr Alex. M c Aulay for having pointed out in his book on Octonions, p. 251, 

 that I had overlooked this exception when enunciating the Theorem of the reduced wrench in the 

 Theory of Screws (1876). 



