102 Pear-shaped Configurations of Equilibrium [CH. v 



103. The fact that the pear-shaped series is initially unstable shews that 

 a rotating mass cannot evolve by slow secular- changes through a series of 

 pear-shaped figures. This somewhat diminishes the interest of the pear- 

 shaped series in the problem of cosmogony, but nevertheless it remains im- 

 portant to obtain as clear an idea as we can of the nature of this series. For 

 we shall find, when we come to the discussion of dynamical motions, that the 

 unstable series are of the utmost importance in directing the course of dy- 

 namical or cataclysmal motions such as occur when statical evolution is no 

 longer possible. 



There is nothing in abstract theory to prevent us following out the con- 

 figuration of the pear-shaped series as far as we like, but the labour of 

 computation would be so great as to make this course impracticable. 



A problem which admits of very much easier solution is the two-dimensional 

 problem of tracing out the sequence of configurations of a rotating cylinder 

 of liquid. So far as the three-dimensional case has been solved, the analogy 

 between the two-dimensional and three-dimensional cases is so very close 

 that we may reasonably hope that it will persist beyond. If this is so, we can 

 discover the general nature of the solution to the three-dimensional problem 

 by examining that of the much simpler two-dimensional problem. We ac- 

 cordingly turn to a discussion of the two-dimensional problem. 



THE CONFIGURATIONS OF EQUILIBRIUM OF ROTATING 

 LIQUID CYLINDERS 



104. Let F (x, y) = be the equation of a cylindrical boundary in the 

 plane of x, y, and for simplicity let us assume the axis of x to be one of 

 symmetry. 



Let us change to complex variables f, 77 defined by 



and let the equation of the curve become 



/(f,i?) = .............................. (287). 



If the original curve was symmetrical about the axis of x, the function/ 

 must of course be symmetrical in f and 77. 



To write down the potential of a homogeneous cylindrical mass having 

 (287) as the equation of its cross-section, we solve the equation explicitly 

 for f ; let the solution be 



f-*(i)M(l) + *(>) ..................... (288), 



where <f>(rj) and -^(77) are terms in ascending and descending powers of 77 

 respectively, say 



........................ (289), 



+: ..... .................. (290). 



