'}• 



Motion of a Perforated Solid in Liquid. 349 



Since these equations do not involve u or p, we see that 

 no forces will have to act on the solid in order to 'maintain a 

 screw motion whose axis coincides with the central axis of the 

 impulse. 



13. To interpret the equations still further, let us suppose 

 that u and p are both zero, since they do not enter into the 

 equations of motion. Then the motion whose components are 

 (0, v, w, 0, q, r) consists of two screws whose axes are the axes 

 of y and z respectively, and, by the theory of screws, these 

 are equivalent to a single screw whose axis is a certain 

 straight line intersecting the axis of x and perpendicular to 

 it. We may take this straight line as our axis of 0, for 

 hitherto we have only fixed the position of the axis of x. 

 We have then 



v=0, 2=0. 



The equations (34) therefore reduce to 



X=0, L=0, 



Y=rB, M.=wB + rA, £> . . . (35) 



Z=0, N=0. 



Hence the solid is acted on by a wrench (Y, M) whose axis 

 is the axis of y. Thus the axis of the impressed wrench is 

 perpendicular to the central axis of the impulse of the 

 fluid motion, and to the axis of the screw motion of the body. 



Let n be the pitch of the impulse, vr the pitch of the 



screw motion of the solid, P the pitch of the impressed 



wrench, then 



n _A w M 



11-g, *r_-, .t-y> 



and therefore by (35), 



p=^+n ....... (36) 



is the relation connecting the three pitches. 



In particular, if r = the equations of motion give 



Z = 0, M=mS, 



showing that a couple M about the axis of y will produce 

 translational motion with velocity M/S along the axis of z. 



14. More generally, let the motion be a screw motion 

 about an axis whose inclination to the axis of as is 6 and 

 whose shortest distance from that axis is a. Take this 

 shortest distance as the axis of y, and let the screw motion 

 consist of a linear velocity V combined with an angular 

 velocity O, the pitch V/I2 being denoted, as before, by «■. 



Phil. Mag. S. 5. Vol. 35. No. 215. April 1893. 2 B 



