178 Prof. G. H. Darwin. Stability of the Pear-shaped [June 10, 



nion function, and this function is employed in the investigation of 

 the properties of a four-system of linear transformations, of the 

 general quadratic transformation, and of the non-linear one-to-one 

 correspondence of points in space. The method of quaternion 

 arrays* is applied to the discussion of ^-systems of linear transforma- 

 tions, and of the critical assemblages of points, lines and planes con- 

 nected with each system of transformations. Finally, in the con- 

 cluding section it is explained how the method of the paper may he 

 applied to hyper-space, or to the discussion of functions of any 

 number of variables ; and in many cases the formula?, obtained in the 

 course of the paper with special reference to three dimensions require 

 no modification to fit them for the general case of n variables. 



" The Stability of the Pear-shaped Figure of Equilibrium of a 

 Eotating Mass of Liquid." By G. H. Dae win, F.E.S 

 Plumian Professor and Fellow of Trinity College, in the 

 University of Cambridge. Pteceived and Bead June 19 

 1902. 



(Abstract.) 



At the end of a previous paperf it was stated that the stability of 

 the pear-shaped figure could not be definitely proved without further 

 approximation. After some correspondence with M. Poincare during 

 the course of my work on that paper, I attempted to carry out the 

 second approximation, but found myself foiled at a certain stage of 

 the work. Meanwhile he had turned his attention to the subject, and 

 he has; shown how the difficulty wh.ch stopped me may be overcome. 

 He has not, however, pursued the arduous task of converting his 

 results into numbers, so that he leaves the question of stability un- 

 answered. 



M. Poincare was so kind as to allow me to detain his manuscript on 

 its way to the Kdyal Society for a few days, and being thus assisted I 

 was able to resume my attempt under favourable conditions, and this 

 paper is the result. The substance of the analysis of this paper is, of 

 course, essentially the same as his, but the two present but little super- 

 ficial resemblance. It is perhaps well that the two investigations of so 

 complicated a subject should be nearly independent of one another. 



If a mass of liquid be rotating like a rigid body with uniform 

 angular velocity, the determination of the figure of equilibrium may be 



* 6 Trans. R. Irish Acad., vol. 32, pp. 17—30. 

 f 'Phil. Trans.,' A, vol. 198, pp. 301—331. 

 X ' Phil. Trans.,' A, vol. 198, pp. 333—373. 



