THE PRINCIPAL SCREWS OF INERTIA. 59 



The co-ordinates of a scrciv belonging to a given screw 

 complex are simplified by taking n co-reciprocal screws 

 belonging" to the given screw complex as a portion of 

 the six screws of reference. In this case, out of the six 

 co ordinates ai, . . . . a 6 of a screw a, which belongs to 

 the complex, 6-n are actually zero. Thus we are en- 

 abled to give a more general definition of screw co- 

 ordinates, which will apply to a screw-complex of every 

 order from i to 6, both inclusive. 



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

 screw a, which belongs to a certain screw complex of the 

 n th order, be decomposed into n wrenches of intensities 

 ai, . . . . a* on n co-reciprocal screws belonging to the 

 same screw complex, then the n quantities a t , .... a,, 

 are said to be the co-ordinates of the screw a. Thus the 

 pitch of a will be represented by / t a? + . . . + / n a n 2 . The 

 virtual coefficient of a and |3 will be 2 (/iaj3i + . . . +/ B aj3 n ) 



We may here remark that one screw can always be 

 found upon a screw complex of the n th order reciprocal 

 to n - i screws of the same complex. For, take 6 % - n 

 screws of the reciprocal screw complex, then the required 

 screw is reciprocal to 6 - n + n - i = 5 known screws, and 

 is therefore determined ( 27). 



66. The Reduced Wrench. A wrench which acts upon 

 a constrained rigid body may always be replaced by a 

 wrench on a screw belonging to the screw complex, 

 which defines the freedom of the body. 



Take n screws from the screw complex of the n th 

 order which defines the freedom, and 6 - n screws from 

 the reciprocal complex. Decompose the given wrench 

 into components on these six screws. The component 

 wrenches on the reciprocal complex are neutralized by 

 the reactions of the constraints, and may be discarded, 

 while the remainder must compound into a wrench on 

 the given screw complex. 



