844 On the Motion of Solid Bodies through a Liquid, 



instability]. Compare Tait and Steele's 'Dynamics of a Par- 

 ticle/ § 149 (15), Species IV., case A=0 and AB finite; also 

 limiting variety between Species I. and Species V. The glo- 

 bule will describe the same circle in the opposite direction if 

 projected with the same velocity opposite to that of the fluid. 

 If the globule be projected either in the direction of the liquid's 

 motion or opposite to it with a velocity less than that of the 

 liquid, it will move along the Cotesian spiral (Species I. of Tait 

 and Steele) from apse to centre in a finite time with an infinite 

 number of turns. If it be projected in either of those directions 

 with a velocity greater by v than that of the liquid, it will move 

 along the Cotesian spiral (Species V. of Tait and Steele) from 

 apse to asymptote. Its velocity along the asymptote at an infi- 

 nite distance from the axis will be 



\A(' + s)' 



and the distance of the asymptote from the axis will be 



4- - 

 2ira 



where a denotes the distance of the apse from the axis, and - — 



r 27ra 



the velocity of the liquid at that distance from the axis. If the 



globule be projected from any point in the direction of any 



straight line whose shortest distance from the axis is p } it will 



be drawn into the vortex or escape from it, according as the 



component velocity in the plane perpendicular to the axis is less 



or greater than = — . It is to be remarked that, in every case in 



which the globule is drawn in to the axis (except the extreme 

 one in which its velocity is infinitely littJe less than that of the 

 fluid, and its spiral path infinitely nearly perpendicular to the 

 radius vector), the spiral by which it approaches, although it has 

 always an infinite number of convolutions, is of finite length, 

 and therefore, of course, the time taken to reach the axis is finite. 

 Considering, for simplicity, motion in a plane perpendicular to 

 the axis, at any point infinitely distant from the axis let the glo- 

 bule be projected with a velocity v along a line passing at 

 distance p on either side of the axis. Then if t denote the ve- 

 locity of the fluid at distance unity from the axis [which is equal 



K 

 to o""]; ailC ^ *f We P Ut 2 



*~ l r.fr («) 



