222 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



XYZ L'M'N' denote the same for a point 0' whose coordinates 

 Then 



.37) M=M' + 



N=N' + xY-yX. 



In order that the point O 1 may lie on the central axis, the direction 

 of resultant and couple must coincide , or 



*L = M- = *[L 

 X~ Y ~ Z ' 



hence the equations of the central axis in Cartesian coordinates are 



L-yZ-\-zY_ M-zX + xZ N-xY+yX 

 ~^~ ~Y~ ~W~ 



The equation of the focal or polar plane to a point x' y r z r is, 

 since it is perpendicular to L'M'N', 



39) (x-x')L' + (y-y')M' + (2-z')N' = 



and inserting the values of L'M'N', 



or, more symmetrically arranged, 



40) L(x-x') + M(y-y') + JV (*-*') 



+ Y(xe' - 0x') + Z(yx' - xy') = 0. 



This equation is symmetrical with respect to xyg, x'y'z', hence if 

 x' y' z' is fixed, xyg is on its polar plane, or if xyz is considered 

 fixed, x' y' &' is on its polar plane, showing the reciprocal relation 

 of pole and polar. 



If the vector system is to reduce to a single vector, the resultant 

 and couple at any point must be perpendicular, or 



41) LX + MY+NZ = 0. 



We must have in general, at any point, $cos# = $ that is, 



42) 





and the pitch p is given by 



4 QN P 3, LX+MY+NZ 



2^t == ~R = ~"X 8 + r*-f Z 2 



The volume of the tetrahedron on a vector -cross is 



44) I ES, = \(LX+ MY+ NZ), 



and this, like the last expression, is independent of the choice of 

 origin or axes, that is, is an invariant. 



