52 BEPOET — 1882. 



supposition that the plane docs not exist, and then adds the motion 

 obtained by impressing on every point of the plane velocities equal and 

 opposite to those at the same point produced by the motion of the sphere, 

 and again, takes account of the ' reflected ' motion of this from the surface 

 of the sphere. If the fourth powers of the ratio of the radius to the 

 distance from the plane be neglected he found that the mass of the 

 equivalent solid is i(l + Sa^/Sh^) times the mass of fluid displaced. When 

 the sphere moves parallel to the plane, the problem is treated by supposing 

 the plane removed, and an equal sphere to move as the optical image of 

 the first. Here under the same circumstances the mass of the equivalent 

 solid is -5(1 + oa^ jlGP) times the mass of the fluid displaced. In both 

 cases h denotes the distance of the centre from the plane. His discovery 

 of the image of a doublet whose axis goes through the centre of the sphere 

 enabled him to solve further problems ' which, however, he did not 

 publish at the time. One of them is printed in the first volume of his 

 collected works and is referred to below. The fact that a sphere projected 

 from the bounding plane moves as if accelerated, whilst if it is projected 

 parallel to the plane it moves as if attracted to it, was deduced from 

 general reasoning in Thomson andTait's ' Natural Philosophy,' published 

 in 1866. 



The whole question of the general motion of two or more spheres has 

 been considered very fully by Bjerknes in a series of papers dating from 

 18G8 onwards. In his first paper ' he treats of the movement of two 

 spheres moving in their line of centres, and shows how a series for the 

 velocity-potential may be obtained. He investigates more particularly 

 the case when their distances are so great that inverse powers of the 

 distance greater than the seventh may be neglected, and shows that the 

 uniform motion of one sphere, along the line of centres produces an 

 apparent repulsive force towards the centre of the other : also the general 

 movement of several spheres whose distances from each are so large com- 

 pared with their radii, that inverse powers of this ratio greater than the 

 fourtb can be neglected. Under these circumstances the action between 

 any two spheres is independent of the presence of the others. Forces occur 

 depending on the acceleration and the kinetic energy of the spheres, that due 

 to acceleration varying inversely as the third power of the distance, whilst 

 that due to the square of the velocity depends on the inverse fourth power. 



In 1871 Guthrie 3 published a curious theorem, due to Thomson, 

 Thomson found that if two spheres are in fluid and oscillating along the 

 line joining their centres, then, if the density of one of the spheres is less 



' In the introduction to Lis p.aper on ' The internal friction of fluids on the motion 

 of Pendulums' {Trana. Camb. Phil. Soc. ix. 1850), he says, speaking of this discovery : 

 ' It enabled mc to calculate the resistance to a sphere oscillating in presence of a 

 fixed sphere or within a spherical envelope, or the resistance to a pair of spheres 

 either in contact, or connected by a narrow rod, the direction of oscillation being, in 

 all these oases, that of the line joining the centres of the spheres. . . . Tlie method 

 even applies, as Professor Thomson pointed out to me, to the uncouth solid bounded 

 by the exterior segments of two intersecting spheres, provided the exterior angle of 

 intersection be a snbmultiple of two riglit angles.' 



- ' Om den samtidige Bevaelgelse af kugleformige Legemer i et inkompressibelt 

 Fluidum.'— i^)r/(«Mf/. iikand. Naturfors. Christiania (1868). 



' ' On approach caused by Vibration,' Phil. Mag. xli. (4) p. 423. There is clearly a 

 printer's error in the result. In the formula the fifth root occurs ; in the numerical 

 example the third root. I have ventured in the text to svibstitute the correct value, 

 viz. the fourth. This paper of Guthrie's contains experiments on the action between 

 bodies moving in fluids and also references to the work of others, For fxuther notices 

 see below. 



