74 REPORT — 1881. 



distance on the other side it will have its path turned towards or from 

 the fixed obstacle according as the direction of the impulse is in the same 

 direction as, or the opposite to, that of translation. This follows at once 

 after it is proved that the effect of the fixed obstacle is to add an impul- 

 sive component towards itself. In the paper which treats of polycyclic 

 motion with several solids is given an example of a sphere moving in a 

 fluid in which there are infinitely fine immovable curves round which 

 polycyclic motion exists. Here the sphere moves as a material particle, 

 whose mass is equal to the mass of the sphere and half the fluid dis- 

 placed, under the action of the impressed forces, and in a field of force 

 whose potential is W, where W is the work done in moving the sphere to 

 an infinite distance from the cores of the polycylic motions ; or the 

 difference of the fluid energies in the cases when the sphere is not 

 present, and when it is. When the core is an infinite sti-aight line, the 

 paths of globules arbitrarily projected are Cotes's spirals. 



I have thought it well to refer to those papers of Thomson's together, 

 as they form a connected series, although between their publications im- 

 portant essays have appeared from other investigators. The first of these 

 requiring mention is Kirchhoff, who taking up the question of the general 

 motion of a body of revolution as treated in Thomson and Tait applied 

 the same methods to extend their results when the motion is not con- 

 fined to one plane. His investigation^ was published in March, 1870. 

 He deduced by analysis alone the Eulerian^ form of the equations of 

 motion in the same form as Thomson's. Taking the origin at the centre 

 of reaction he obtains equations for the velocities and the co-ordinates 

 which enable them to be expressed as elliptic functions of the time. 

 Considering more closely the case already solved in Thomson and Tait's 

 treatise, he finds exphcitly the velocities in terms of the time, and ex- 

 presses the constants in terms of the constants of the kinetic energy and 

 the initial motion. Two cases pi'esent themselves which dej^end on the 

 energy-constants, and each of these subdivides into two subcases accord- 

 ing to the initial motion. These may be expressed thus. If it requires 

 a less impulse to produce unit velocity along the axis than ijeiyeiuUcular 

 to it, then the velocity along the former is expressible by means of the 

 sn functions and vice versa,. Calling the larger and smaller impulses A, B 

 respectively, each case divides into two subcases, according as the ratio 

 of the energy due to rotation bears to the energy due to the other velocity 

 a greater or less ratio than (A — B)/B which corresponds to the case of 

 the rotation and velocity being expressed by dn, en respectively, or vice 

 versa. Kirchhoff finds also that it is possible to project a solid of revolu- 

 tion with rotation round a line perpendicular to its axis, so as to describe 

 a cu'cular helix — a result, which by Thomson's theory of the impulse, is 

 at once seen to be true. In the same volume Kirchhoff^ considered the 

 forces between any two infinitely thin rings with circular section through 

 which cyclic motion is taking j^lace, and proved that the apparent forces 

 between them are equal to those which would exist between them if they 

 were conductors and electric currents flowed along them. The same 



■ ' Ueber die Beweguug eiues Kotationskorpers in einer Fliissigkcit,' Borch., Ixxi. 

 p. 237. 



2 Thomson has proposed to keep the term ' Eulerian ' for equations of motion 

 referred to axes fixed in the moving body. 



^ ' Ueber die Kriif te, welche zwei unendlich diinne starre Einge in einer Fliissigkeit 

 scheinbar auf einander ausiiben konnen,' Boreh., Ixxi. p. 26'^ 



