74 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND 



analogous situation arises in considering the directions in which lines of force start out 

 from a point of equilibrium in an electrostatic field.* 



37. A more interesting illustration of the difficulty will be found in an investigation 

 of the figures of equilibrium of rotating liquid cylinders which I published in 1902.1 

 In this paper the equation of the cross-section of a figure of equilibrium corresponding 

 to a rotation w is supposed to be expressed in the formj 



(162) 

 '&irpl 



where f,, . ... are functions of >j ; g, ; are complex co-ordinates given by 

 = x + iy, >i = xiy, and x, y are ordinary Cartesian co-ordinates measured from the 

 axis of rotation. The quantity is a parameter, analogous to the e of the present 

 paper, measuring distance from the point of bifurcation at which the pear-shaped figure 



2 



and the elliptic cylinder coalesce. At this point of bifurcation, 1 ^ = f , so that 

 = of which the value is shown to be 



* If V is the potential of ,in electrostatic field, the equation of a line of force will be 



a a a 



T 



where /, m, n are direction-cosines. Two lines of force will meet in a point of equilibrium (just as two 

 linear series meet in a point of bifurcation), and the condition for this is 



BV av av 



= 5- = ............. (n.) 



Cx cy oz 



Let ;>,(,, //, : l)e a point of equilibrium satisfying (ii.), then, if e is a small quantity of the first order, the 

 point 



ii + Ac, ;// + [M, .r + ve 



will, as e varies from zero upwards, trace out a line passing through :, i/ , % The condition that this 

 shall be a line of force is, as far as first powers of r, 



cV oV 8V 



and this is satisfied (analytically) because of equations (ii.) for all values of A, /j,, v. Thus, as far as first 

 powers of c, there are as many lines of force through the point of equilibrium as there are values of 

 A, p, v ; an infinite number. But on going as far as -, it becomes clear that there are only t\vo true lines 

 of force through this point. The condition that a point of equilibrium shall exist is also the condition 

 that, if the approximations are not carried far enough, there shall be the confusion of an infinite number 

 of lines appearing to satisfy the condition for a line of force, and the analogous statement is true for 

 points of bifurcation and linear series of figures of equilibrium. 



t " On the Equilibrium of Rotating Liquid Cylinders," ' Phil. Trans.,' A, 200, p. 67. 



\ Loc. cit., equation (71), p. 86 



