, 115.] SPECIAL CASES OF MOTION. 129 



There are therefore three permanent screw-motions such that the 

 corresponding impulsive wrench in each case reduces to a couple 

 only. The axes of these three screws are mutually at right angles, 

 but do not in general intersect. 



115. We will now shew that in all cases where the impulse 

 consists of a couple only, the motion can be completely determined. 

 It is convenient, retaining the same directions of the axes as before, 

 to change the position of the origin. To transfer the origin to 

 any point (x, y, z) we must write 



u + ry qz, v + ps r.v, w -f qx py 



for u, v, w, respectively. We have then in the expression for the 

 kinetic energy 



new 1T= - Bx + M&quot;, new N = Cx + A r/ , &c., 

 so that if we make 



.- -- - 



- &quot;~B ~C&amp;gt; -- C A ~A B&quot; 



we have, in the new expression for 2T, 



jr_jr ^ = // Jtf 



B == C C ~ A A~ B 



Let us denote the values of these pairs of equal quantities by 

 a, /3, 7 respectively. The formulae (26) may then be written 



where 



The motion of the body at any instant may be conceived as made 

 up of two parts ; a motion of translation equal to that of the 

 origin, and one of rotation about an instantaneous axis passing 

 through the origin. The &quot;latter part is to be determined by the 

 equations 



d\ 8 . 



_ =rp-qv, &c., &c., 



d c?&amp;lt;=) dft dS 



or Ti-7- = r -3 -- ?j&amp;gt; *^i ^&quot; c - 



dt dp dq &quot; dr 



