128 ON THE MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. V. 



There exists then for every body, under the circumstances here 

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



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

 manent translation are those in which the impulse reduces to a 

 couple only. The equations (20) or (24) are satisfied byf, 77, f =0, 

 and X, ft, v constant, provided 



\ l*&amp;gt; v , . . 



-=-=-, = &, say.. ................... (25). 



p q r 



If the axes of co-ordinates have the special directions adopted in 

 Art. 113, the conditions f, 77, =0 give us at once u, v, w in terms 

 of p, q, r, viz. 



tt = -^ + Z / + rV ,&c.,&c ........... (26). 



xL 



Substituting these values in the expressions for X, //,, v obtained 

 from (23), we find 



d d d 



where 



2 = ^p* + &amp;lt;&q* + Wir 2 + 2^ qr + m rp + m pq ..... (28); 

 the coefficients in this expression being determined by the formulae 



&quot; 



-_ 



A B G 



LL&quot; MM&quot; N N&quot; 



A B C 



&c., &c. 



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

 of the impulse is zero. Introducing the conditions (25) for steady 

 motion, we have to determine p : q : r the three equations 



(29). 



The form of (29) shews that the line whose direction-ratios are 

 p : q : r is parallel to one of the principal axes of the ellipsoid 



8 (a?, y, z) = const ..................... (30). 



