224 VI. SYSTEMS OF VECTORS. DISTftlBUT. OF MASS. INSTANT. MOTION. 



are the coordinates of a line (not of a general system of vectors), 

 that is fulfill the relation 



If they do, then every line of the complex cuts the line X Y ,Z OJ L Jf ./V , 

 and the equation may be considered the equation in Pliicker's co- 

 ordinates of the line X Q Y Q Z Q L MoN Q (see Clebsch, Geometric, Yol. II, 

 p. 51). For further information on this subject, the reader may 

 consult, Ball, Theory of 



67. Momentum Screw. Dynamics. The previous sections 

 have shown how to combine systems of vectors having different 

 points of application, provided they are unchanged if slid along their 

 lines of direction. As one particular system to which the operation 

 is applicable we have had the various rotation - velocities of 

 a rigid body, as another, sets of forces applied to a rigid body. 

 That these vectors are susceptible of such treatment may be considered 

 as due to properties of a rigid body, rather than of the vectors 

 themselves. We have however previously dealt with two other sorts 

 of vectors which may be dealt with in similar fashion, on account 

 of their physical nature, and independently of the nature of the 

 bodies in which their points of application lie. By means of these 

 properties we are able to connect the kinematical aspect of a rigid 

 body, as expressed by its instantaneous screw motion, with its 

 dynamical aspect, as expressed by an applied wrench about another 

 screw. 



If for each point of the system we consider the momentum, 

 whose six coordinates (one being redundant), in the sense of 66 are, 



mv x , mv y , mv z , m(yv z 2Vy), m(zv x xv z ), m(xv y yv x ), 



and form the general resultant, we obtain a system whose co- 

 ordinates are 



M x = Zmv x , H x = Zm (yv z zv y }, 



48) My = 2mvy, H y = 2m (zv x xv z }, 



M z = 2mv ZJ H z = Urn (xv y yv x ), 



which represent the momentum of the system, the three projections 

 M x , My, MS, being more particularly characterized as the linear 

 momentum, the others H x , H y , H z , as the angular momentum or 

 moment of momentum with respect to the origin. 



We have now by the general principles of dynamics, as shown 

 in 32, 45), 33, 61), the fact that the time -derivatives of these 

 six components of momentum are equal to the corresponding com- 

 ponents of the resultant wrench, 



