IlECENT PROGJEESS IN HYDRODYNAMICS. 77 



sets of screws just mentioned do not in general intersect, but their axes 

 lie along the alternate edges of a parallelopiped. Take the centre of 

 this parallelopiped, and call it O. Then the motion of the body is 

 compounded of the motion of O, and a motion about 0. Describe 

 about 0, as centre, a certain ellipsoid, which, as in Poinsot's 

 representation, gives the motion relative to by rolling on a 

 plane with angular velocity proportional to the instantaneous axis 01. 

 The motion of is then represented by drawing round another deter- 

 minate quadric. Suppose 01 cuts the quadric in P, and OM is the 

 perpendicular from on the tangent plane at P ; then the motion is 

 completely represented by supposing the Poinsot ellipsoid and plane to 

 move bodily in the direction of OM with a velocity proportional to 

 01 / (OP. OM). For particular forms of the body this naturally 

 simplifies very much; for instance, in the case of an isotropic helicoid, 

 any screw through O is a permanent one. The motion is stable about 

 two of these fundamental screws and unstable about the third. Some of 

 Lamb's results have since been obtained by Craig.' The steady motion 

 of a solid of revolution has also been treated by Greenhill,- who has given 

 an expression for the least rotation about the axis of a prolate solid of 

 revolution that it may keep its point in front. An investigation similar 

 to Greenhill's was given by Craig ^ about the same time. 



The general theory of the motion of more than one body in a fluid has 

 not hitherto received much attention. Many special problems have been 

 solved, especially the case of two or more spheres by several writers. But 

 these, beyond those referred to above, will best be noticed later under 

 special problems. Most of the qualitative results obtained for spheres, no 

 doubt hold good for solids in general. We may then expect that bodies 

 vibrating in a fluid will appear to act on one another with forces varying 

 according to inverse powers of the distance higher than the second, while 

 pulsating bodies (or bodies changing their volume periodically) will have 

 the chief part of their action proportional to the inverse square of the 

 distance. 



(e) Viscous Fluids. 



The general theory of viscous fluids presents diSicuIties which can 

 scarcely even yet be said to be settled. The equations of motion have not 

 the same degree of certainty as in the case of perfect fluids, partly on 

 account of the difiiculty of finding a satisfactory theory on which friction 

 is to be explained, and partly on the difficulty, as Stokes has pointed out, 

 of connecting the oblique pressures on a small area with the difl^erentials 

 of the velocities. In the last report to the Association Professor Stokes 

 has given a clear description of the various methods by which, up to 1846, 

 Navier, Poisson, St. Venant, and he himself had attacked the problem. 

 Since then several others have investigated the subject with results most 

 of which can be expressed in (what may be called) the typical form — 



where denotes the cubical compression. In Stokes' form, which is that 



' ' The motion of a solid in a fluid,' Amer Jour. Math., ii. p. 107. 



' ' Motion of an ellipsoid in fluid,' Quart. Jour. Math., xvi. p. 255, and xvii. p. 8C ; 

 also for numerical applications to gunnery, see papers by the same author in the 

 Proc. Royal Artillery Inst. x. xi. ; and art. ' Hydrodynamics,' Encyo. Brit., 9th edit. 



' ' On the motion of an ellipsoid in fluid,' Amer. Jour. Math. ii. p. 271, 



