121-122] STEADY MOTIONS. 179 



twist about a certain screw; and the condition that this motion 

 should be permanent is that it should not affect the configuration 

 of the impulse (which is fixed in space) relatively to the body. 

 This requires that the axes of the screw and of the corresponding 

 impulsive wrench should coincide. Since the general equations 

 of a straight line involve four independent constants, this gives four 

 linear relations to be satisfied by the five ratios u : v : w : p : q : r. 

 There exists then for every body, under the circumstances here 

 considered, a singly-infinite system of possible steady motions. 



Of these the next in importance to the three motions of permanent 

 translation are those in which the impulse reduces to a couple. The equa 

 tions (1) of Art. 117 are satisfied by , 77, =0, and X, /u, v constant, provided 



*/P = P-l ( l = v /r, =, say ........................... (i). 



If the axes of coordinates have the special directions referred to in the 

 preceding Art., the conditions , 77, =0 give us at once w, v, w in terms 

 of p, q, r, viz. 



* (Lp + L q+ L&quot;r)IA,\ 



v=-(Mp + M q + M&quot;r)/B\ .......................... (ii). 



w= - (Np+N q +N&quot;r)/C ) 



Substituting these values in the expressions for X, p, v obtained from Art. 

 119 (3), we find 



. &amp;lt;tfe dQ dQ 



X = dp ^ = ^ &quot; = ^ ........................... ( m )&amp;gt; 



where 26 (p, q, r) = %? 2 + &amp;lt;% 2 -f Hr 2 + Vfflqr + ZQ&rp + 2R pq ......... (iv) ; 



the coefficients in this expression being determined by formulae of the types 



L L&quot; M M&quot; N N&quot; 



&quot; ~A ~^r ~c~ 



These formulae hold for any case in which the force- constituent of the impulse 

 is zero. Introducing the conditions (i) of steady motion, the ratios p : q : r 

 are to be determined from the three equations 



The form of these shews that the line whose direction-ratios are p : q : r 

 must be parallel to one of the principal axes of the ellipsoid 



0(.r, y, 2) = const ............................... (vii). 



122 



