65, 66] RECIPROCAL SCREWS. 



But when E f is changed to the origin of R k it gives rise to a moment 

 perpendicular to E f equal to E f d, d being the perpendicular distance 

 between the screws. This moment therefore makes with E k the 



angle a -j- - g -> and the work done by it in the rotation E k is 



dE f E k cos a -f y = dE f E k sin . 

 Thus the whole work is 

 34) W = E f E k ( cos a - d sin a . 



It is symmetrical with respect to both screws, hence the wrench and 

 twist might have been interchanged. 



The geometrical quantity in parentheses is called the virtual 

 coefficient of the two screws, and if it vanishes no work is done, 

 that is, a body free to twist only about a particular screw is in 

 equilibrium under a wrench about another screw if the virtual coef- 

 ficient of the two screws is zero. The two screws are then said to 

 be reciprocal. 



66. Analytical Representation. Line Coordinates. In 



Pliicker's line coordinates referred to any origin, since each component 

 of vector does work on the corresponding component of couple in 

 the other system, 



35) W = X f L k + Y f M k -f Z f N k + L f X k + M f Y k + N f Z k . 



If a screw is reciprocal to two screws on a cylindroid, it is 

 evidently reciprocal to all the screws on it. 



For two screws to be reciprocal, the condition is, 



36) X,L 2 + Y,M, + Z,N 2 + L,X 2 + M,Y 2 + N,Z, = 0. 



If the coordinates of one of the screws be constant, while those of 

 the other be variable, this is the equation 18) of a complex of the 

 first degree, so that all the screws reciprocal to a given screw form 

 such a complex. 



Since between the six coordinates X 1 Y 1 Z 1 L 1 M 1 N 1 there is 

 always the identical relation 



X, A +Y 1 M 1 + Z 1 N 1 = 0, 



we may always make them satisfy five equations like the above, 

 that is, we may always find a screw reciprocal to five arbitrarily 

 given screws. 



Suppose the coordinates of the system of vectors for an origin 

 are XYZLMN, being the projections of E and 8 at 0. Let 



