414 THE THEORY OF SCREWS. [377, 



377. Different Screws on the same axis. 



Let the body be displaced from a standard position to another position 

 denned by the co-ordinates #/, 2 , . . . 6 , and let it then be set in rotation 

 about a screw of zero pitch with a twist velocity whose co-ordinates are 

 lt 2 , ... 6 . Let the kinetic energy of the body in this condition be T. 



Suppose that in addition to the rotation about 6 the body of mass M 

 also received a velocity v of translation parallel to 0. Then the kinetic 

 energy of the body would be T , where 



It is obvious that the position of the body, i.e. the co-ordinates $/, 0.f, ... e , 

 can have no concern in ^Mv 2 , whence 



dT dT 

 , a , = -J0-, , and similar equations. 



at/i CtC/! 



But a body rotating about with an angular velocity and translated 

 parallel to with the velocity v is really rotating about a screw on the 

 same axis as and with a pitch v -f- p. As v may have any value we obtain 

 the following theorem : 



All instantaneous screws lying on the same axis have the same restraining 

 screw. 



378. Co-ordinates of the Restraining Wrench for a free rigid 

 body. 



Suppose the body to have a standard position from which we displace it 

 by small twists #/, . . . 6 around the six principal screws of inertia. While 

 the body is in its new position it receives a twist velocity of which the 

 co-ordinates relatively to the six principal screws of inertia are lt ... 6 . 



To compute the kinetic energy we proceed as follows : Let a point lie 

 initially at oc, y, z, then, by the placing of the body at the starting position 

 the point is moved to X, Y, Z, where 



X = a (0, - 2 ) + y (0 5 f + 6 ) - z (0 3 f + 0/) + x, 

 Y = b (0 3 f - 0/) + z (0, + 2 ) - x (0, + U ) + y, 

 Z = c (0 r ; - 6 ) + x (0, + 0/) - y (0/ + a ) + z, 



in which a, b, c are the radii of gyration on the principal axes. The six 

 principal screws of inertia lie, of course, two by two on each of the three 

 principal axes, with pitches + a, a on the first, + b, b on the second, 

 and + c, c on the third. 



In consequence of the twist velocity with the components #1,... 6 , each 

 point X, Y, Z receives a velocity of which the components are 



