1902.] The Equilibrium of Rotating Liquid Cylinders. 

 we transform by the substitution 



47 



£ = re 19 , 



1 



= re~ 



IB 



and attempt to solve the resulting equation explicitly for f in the 

 form 



this solution being such that the right hand gives the true value of £ 

 at every point of the surface given by equation (ii). The condition 

 that the surface shall be an equilibrium surface under a rotation w is 

 found to be given by the system of equations 



The constancy of area of the curve (ii) can be effected by keeping c\ 

 constant. This method is subject to certain modifications, owing to 

 the possibility of the various series becoming divergent. 



The linear series of circles and ellipses (corresponding to the Mac- 

 laurin spheroids and Jacobian ellipsoids) are investigated without 

 difficulty, and the points of bifurcation on these series are found. The 

 first point of bifurcation on the latter series is shown to lead to a pear- 

 shaped curve, similar to that of Poincare, and it is shown that an 

 exchange of stabilities takes place at this point. 



The linear series of which this pear-shaped figure is the starting 

 point can now be investigated, the equation being expanded in an 

 ascending series of powers of a parameter 6. Since the equations are 

 not linear, the calculation of terms multiplying high powers of 9 is 

 •extremely laborious. The series is, therefore, calculated only as far as 

 6 5 , this being found to give tolerable accuracy so far along the series 

 as the expansion is required. 



After passing through various pear-shaped configurations the fluid 

 is found to assume a shape similar to that of a soda-water bottle with 

 a somewhat rounded end. Beyond this the configuration is found to be 

 suggestive of a tennis-racquet with a very short handle. A "neck" 

 gradually forms at the point at which the handle joins the racquet, 

 and this becomes more pronounced, until ultimately the curve separates 

 into wo parts. 



As we proceed along this series the rotation steadily increases. At 

 the point of bifurcation the value of w 2 /27rp is 0'375 ; when separation 

 takes place this value is about 0*43. It is tolerably clear (although not 

 rigorously proved) that when separation takes place, the primary may 

 be regarded as the Jacobian ellipse, corresponding to rotation 



(iii) 



VOL. LXX. 



0)2/2^ = 0-43 



(iv), 



E 



