77] THE EQUILIBRIUM OF A RIGID BODY. 67 



77. Remark on systems of Linear Equations. 



Let a right line be, as usual, represented by the two equations 

 Ax + By + Cz + D = 0, 



There are here six independent constants involved, while a right line is 

 completely defined by four constants. The fact of course is that these two 

 equations not only determine the right line on which our attention is fixed, 

 but they also determine two planes through that line. Four constants are 

 needed for the straight line and one more for each of the planes, so that there 

 are six constants in all. 



If we are concerned with the straight line only the intrusion of two 

 superfluous constants is often inconvenient. We can remove them by first 

 eliminating y and then x, thus giving the two equations the form 



We have here no more than the four constants P, Q, P , Q , which are indis 

 pensable for the specification of the straight line. 



Of course it may be urged that these equations also represent two planes. 

 No doubt they do, but the equation z = Px + Q is a plane parallel to the axis 

 of y, which is absolutely determined when the straight line is known. The 

 plane Aac+By+Cz + D = Q may represent any one of the pencil of planes 

 which can be drawn through the straight line. 



Analogous considerations arise when the screws of an n-system are 

 represented by a series of linear equations. We commence with the case of 

 the two-system, in which of course the screws are limited to the generators 

 of a cylindroid. 



Let 0j, 2 , ... 6 be the co-ordinates of a screw referred to any six screws 

 of reference. 



Let these co-ordinates satisfy the four linear equations 

 A,e, +A&+ +4 6 = 0, 

 B& + B,0 2 + ... +B,0 B = 0, 



0,6,+ &amp;lt;7 a 2 + ... + c\e c&amp;gt; = o, 

 A0i + A& + ... + A0 8 = o, 



where AI, A, ..., B l , B 2 , ... G ly (7, and D 1} D.,, ... are constants. 



Then it is a fundamental part of the present Theory that the locus so 

 defined is a cylindroid (| 76). 



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