Motion of a Perforated Solid in Liquid. 351 



motion. This impressed wrench may be divided into two 

 parts, one being due to the terms 37 + 3^ in the modified 

 function, the other being due to the terms 



^u + ^v + ^w + Xp + fiq-]- vr. 

 The first portion is the same as if the motion were acyclic, 

 and represents, therefore, the wrench which would have to be 

 impressed on the solid in order to maintain the given motion 

 if there were no circulation. The second part represents the 

 additional wrench which must be applied on account of the 

 circulations, and the equations to determine it are of the forms 

 found above. 



We notice, in particular, that if the solid has any screw 

 motion whose axis coincides with the axis of the impulse of 

 the cyclic fluid motion, the latter wrench vanishes ; so that 

 the forces required to maintain the motion are unaffected by 

 the circulations. In other cases the additional wrench is 

 about an axis perpendicular to the axis of the impulse. This 

 is true whatever be the form of the solid and the number of 

 the circulations ; but, as has already been pointed out, the 

 position of the axis of the impulse relative to the solid is not 

 in general independent of the circulations unless the solid has 

 but a single aperture. 



It is probable that these results might be made to furnish 

 mechanical illustrations of certain physical phenomena ; but 

 with these we are not concerned in the present paper. 



Note on the foregoing Paper, 



Concerning the proper measurement of the impulse of the 

 cyclic motion, a difficulty arises ; for, as Mr. Bryan remarks, 

 this motion cannot be set up from rest by impulses applied to 

 the solid alone. Suppose, however, that we close each perfora- 

 tion by a barrier in the usual way, and let the barriers be acted 

 on by the impulsive pressures K x p } K 2 p, . . . respectively. And 

 instead of these impulsive pressures being due to external 

 forces, suppose that they are due to some immaterial mechanism 

 attached to the solid. In general, an impulsive wrench must 

 act on the solid to keep it at rest, and this wrench is the 

 required impulse of the cyclic motion ; for the only other 

 impulses acting on the system are due to the mutual reactions 

 of the solid and fluid, exerted partly over the surface of the 

 solid and partly through the barriers and attached mechanism, 

 and such mutual reactions cannot affect the impulse. The 

 wrench thus found is of course the same as would be 

 obtained by supposing the impulses on the barriers to be due 

 to external impulsive forces, and compounding with these the 



2B2 



