44 BEPOiiT — 1882. 



initial motion, when one circle has a motion of translation. He shows 

 that in this case, which serves very approximately to determine the small 

 vibration under gravity of the inner circle, the mass of the equivalent 

 solid is (5^ + a.^)/(6^ — a^) times the mass of the flaid displaced ; a, b 

 being the radii of the circles. I ' have discussed the most general motion 

 of two circles, either internal, or external to one another, treating the 

 two cases separately where the circles touch or not. The velocity and 

 potential functions are given for any motion of the two circles, and the 

 following particular cases are considered more in detail : (a) the motion 

 of a pendulum inside a circular case, (/3) of a circle in fluid bounded by a 

 straight line, (y) of two circles rigidly connected, (I) the motion of one 

 when the other is fixed, and lastly (e) some properties of the general 

 motion of two free circles, in all cases without cyclic motion. It may be 

 interesting to give some of the results, which admit of quantitative de- 

 termination in finite terms. If (as in /3) a circle be projected from con- 

 tact with the boundary line, in a direction perpendicular to it, the limiting 

 velocity as it moves off to an infinite distance is increased in the ratio 

 (jTT- + () — 1)- (p + l)""'- If it be projected from any point, the future 

 path will have its concavity turned towards the plane, and will turn 

 round and meet the plane or not, according as the direction of projection 

 makes an angle with the perpendicular to the plane greater or less than 

 a certain angle a, which depends only on the distance from the plane. 

 When the circle is projected from contact, the values of a for densities of 

 the circle 0, 1, 10 are about 41° 22'; 51° 14' ; 70° 15' respectively. If in 

 (ci) the circles are equal, and one is projected directly from contact with 

 the other, the limiting velocity is (^tt"^ + p —1)- (p + 1)"^' times the 

 initial velocity. If it were projected in any way it will move as if attracted 

 on the whole by the fixed circle, the path will have its concavity turned 

 towards it, and will have two asymptotes, whose distances from the 

 centre of the fixed circle are (p + F^^y (p + 1)"- times the apsidal dis- 

 tance, where Po is a certain number depending only on this distance. If, 

 for example, they touch when nearest Pq = ^ tt- — 1. If, on the contrary, 

 both are free to move (as in e), and they are projected so that the whole 

 'momentum' of the system is zero, they move as if they repel one 

 another, and the jDatli of one relatively to the other has its convexity 

 towards that other. If they are equal and touch one another at their 

 nearest distance, the distance of the asymptote of the path of one from 

 the centre of the other is Q tt- + p — ly (p + l)"-x sum of radii. 

 If there is cyclic motion between the circles it is possible to have them 

 moving steadily forward through the fluid, always keeping at the same 

 distance, provided the circles are equal. The discussion^ of this motion 

 shows that when the radii and distance of the centres are given there are 

 two possible relations between the velocity of translation and the relative 

 motion, one in which they are in the same direction between the circles 

 and the other in the opposite. If 2 a be the angle between the two 

 internal tangents to the circles, then when a is not nearly ^ tt, the two 

 ratios of the velocity due to cyclic motion alone, at the point half-way 

 between the cylinders, to the velocity of translation is very approximately 

 2 sec a {+ V (1 — cos a sin* o) — 1} , 



' ' On the motion of two cylinders in a fluid,' Quart. Jour. xvi. pp. 113 and 193, 

 and Proc. Camb. Phil. Soo. iii. p. 227 (1878). 



- ' On the condition of steady motion of two cylinders in a fluid,' (Juart. Jour. 

 xvii. p. 194 (1879). 



