Motion of a Perforated Solid in Liquid. 353 



ordinary investigations given by Prof. Lamb, and, in a less 

 intelligible form, by Basset. 



The equations of motion under finite forces may be deduced 

 by equating the change of momentum in a small time-interval 

 St to the impulse of the impressed forces, taking into account 

 the fact that in the interval St the origin has a displacement 

 of translation (u8t, vSt, ivSt) and the axes have rotational dis- 

 placements {pSt 3 qSt, rBt), so that the final momenta are referred 

 to a different set of axes to the original momenta. 



The mode of forming the equations of motion is given by 

 Prof. Greenhill (Encyclopaedia Britannica, art. "Hydro- 

 mechanics ") for the case of acyclic motion, but it is hardly so 

 obvious why in thus forming the equations of motion of a 

 perforated solid, it is necessary to include in the " impulse " 

 terms representing the components of the wrench applied to 

 the harriers as well as to the solid. We may, however, sup- 

 pose the changes which actually occur in the time St to have 

 been produced as follows : — 



1st. Let the solid and fluid be reduced to rest by an impul- 

 sive wrench applied to the solid, and transmitted to a series of 

 barriers crossing the perforations. The components of this 

 wrench will be found to be 



|?+f, &c.,...|^+\, &c. ... 



o^ op 



2nd. The barriers being rigidly connected with the solid, let 

 the latter receive small displacements whose translational and 

 rotational components are (uSt, vSt, ivSt, pSt, qSt, rSt) and let 

 the solid come to rest in its new position. The fluid will 

 evidently also come to rest, and therefore no impulse will be 

 impressed on the system by this change (as may be otherwise 

 seen by supposing the change to take place very slowly). 



3rd. Let the solid be set in motion with velocity-compo- 

 nents (u + 'dufo't .St, . . ^ p + ~dp/ot -St . . .) referred to the 

 new positions of the axes, and let the circulations tc be started 

 in the new position of the solid by a suitable impulsive wrench 

 applied to the solid and transmitted from it to the barriers. 



Then the impulse of the impressed forces (components 

 'K.St . . ., JjSt . . .) is the resultant of the wrenches required to 

 stop the whole system in the first process and to start it again 

 in the third. 



It is, therefore, that impulse which must be compounded 

 with the total impulse in the initial position in order to 

 obtain the total impulse in the final position. 



Whence Hay ward's equations of motion follow at once (as 



