THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 75 



The calculation of presents no difficulty, and equation (162) as far as only is 

 shown to be the equation of a pear-shaped figure. On calculating 2 its value is found 

 to be of the form (cf. 22 of the " Cylinders " paper), 



180, 2825 4375 _ , , , /48 } 1984^ | 



where S. 2 is analogous to the n" of the present paper ; to be exact the rotation for any 

 value of is supposed given by 



2 



ZTT P ~ 



Again, then, as far as O 2 there is a doubly-infinite series of figures of equilibrium, not 

 two singly-infinite series. 



In this earlier and simpler investigation, it was an easy matter to carry the 

 computations to the third, fourth, and fifth orders of small quantities. It was found 

 that the equations giving , for a figure of equilibrium could not be satisfied so long 

 as S 2 was kept indeterminate, they could only be satisfied for one special value of S 2 , 



namely <J 2 = . After having determined the value of <^ in this way, it was 



448 



possible to investigate the stability, and the pear-shaped cylinder was found in point 

 of fact to be stable. What is important in connection with the present paper is that 

 it was not possible to determine the stability of the pear until after its figure had been 

 determined to the third order of small quantities.* 



38. The work of PoiNCARE can hardly be compared in detail with the investiga- 

 tion of the present paper, because POINCARE tacitly assumes the whole point at 

 issue ; namely, that it is possible to determine the beginning of the pear-shaped series 

 from an investigation of figures of equilibrium going only as far as second-order 

 terms. The work of DARWIN admits of detailed comparison, because DARWIN'S 

 work claims actually to have effected the determination which I am compelled to 

 believe, after my investigation, to be impossible. 



It will be clear that any extra condition in addition to the conditions that the 

 figure should be one of equilibrium will provide an additional equation which will 

 reduce the doubly-infinite series inside the rectangle ABCD down to a singly-infinite 

 series represented by a straight line. For instance, the condition that the angular 

 velocity shall remain constant would reduce the rectangle to the straight line QOQ' ; 

 the condition that the angular momentum should remain constant would reduce it to 



* The previous investigation on cylinders and the present one on three-dimensional bodies follow 

 widely different methods ; the present paper is in no sense a translation of the former from two into three 

 dimensions. The two papers were written at an interval of twelve years, and I had hardly referred to the 

 former paper in writing the present one until after I had encountered the difficulty of not being able to 

 determine the stability from the second-order figure ; I then discovered that precisely the same situation 

 had arisen in my former investigation. 



L 2 



