The pear-shaped figure ;, . , | ; 



but I do not know at what stage of its development it become* 

 unstable. 



Professor Jeans has solved a problem which is of interest 

 as throwing light on the future development of the pear-shaped 

 figure, although it is of a still more ideal character than the one 

 which has been discussed. He imagines an infinitely long circular 

 cylinder of liquid to be in rotation about its central axis. The 

 existence is virtually postulated of a demon who is always occupied 

 in keeping the axis of the cylinder straight, so that Jeans lias only 

 to concern himself with the stability of the form of the section of 

 the cylinder, which as I have said is a circle with the axis of rotation 

 at the centre. He then supposes the liquid forming the cylinder to 

 shrink in diameter, just as we have done, and finds that the speed of 

 rotation must increase so as to keep up the constancy of the rotational 

 momentum. The circularity of section is at first stable, but as the 

 shrinkage proceeds the stability diminishes and at length vanishes. 

 This stage in the process is a form of bifurcation, and the stability 

 passes over to a new series consisting of cylinders which are 

 elliptic in section. The circular cylinders are exactly analogous with 

 our planetary spheroids, and the elliptic ones with the Jacobian 

 ellipsoids. 



Fig. 4. 

 Section of a rotating cylinder of liquid. 



With further shrinkage the elliptic cylinders become unstable, 

 a new form of bifurcation is reached, and the stability passes over 

 to a series of cylinders whose section is pear-shaped. Thus far the 

 analogy is complete between our problem and Jeans's, and in con- 

 sequence of the greater simplicity of the conditions, he is able to 

 carry his investigation further. He finds that the stalk end of the 

 pear-like section continues to protrude more and more, and the 

 flattening between it and the axis of rotation becomes a constriction. 

 Finally the neck breaks and a satellite cylinder is born. Jeans'a 

 figure for an advanced stage of development is shown in Fig. 1. but 



exist with stability for greater rotational momentum. My own work seems to indioata 

 that the opposite is true, and, notwithstanding M. Liapounoff's deservedly great authority, 

 I venture to state the conclusions in accordance with my own work. 



