ADDRESS. 21 
comprises all those various forms which a mass of rotating liquid is capable 
of assuming under the conjoint influences of gravitation and rotation. If 
the earth were formed throughout of a liquid of the same density, it 
would be one of the species of this family ; and indeed these researches 
date back to the time of Newton, who was the first to explain the figures 
of planets. 
The ideal liquid planets we are to consider must be regarded as work- 
ing models of actuality, and inasmuch as the liquid is supposed to be 
incompressible, the conditions depart somewhat widely from those of 
reality. Hence, when the problem has been solved, much uncertainty 
remains as to the extent to which our conclusions will be applicable to 
actual celestial bodies. 
We begin, then, with a rotating liquid planet like the earth, which is 
the first stable species of our family. We next impart in imagination 
more rotation ‘to this planet, and find by mathematical calculation that 
its power of resistance to any sort of disturbance is less than it was. In 
other words, its stability declines with increased rotation, and at length 
we reach a stage at which the stability just vanishes. At this point the 
shape is a transitional one, for it is the beginning of a new species with 
different characteristics from the first, and with a very feeble degree of 
stability or power of persistence. As a still further amount of rotation is 
imparted, the stability of the new species increases to a maximum and 
then declines until a new transitional shape is reached and a new species 
comes into existence. In this way we pass from species to species with 
an ever-increasing amount of rotation. 
The first or planetary species has a circular equator like the earth ; 
the second species has an oval equator, so that it is something like an egg 
spinning on its side on a table; in the third species we find that one of 
the two ends of the egg begins to swell, and that the swelling gradually 
becomes a well-marked protrusion or filament.!_ Finally the filamentous 
protrusion becomes bulbous at its end, and is only joined to the main 
mass of liquid by a gradually thinning neck. The neck at length breaks, 
and we are left with two separated masses which may be called planet 
and satellite. It is fair to state that the actual rupture into two bodies 
is to some extent speculative, since mathematicians have hitherto failed 
to follow the whole process to the end.? 
In this ideal problem the successive transmutations of species are 
brought about by gradual additions to the amount of rotation with which 
the mass of liquid is endowed. It might seem as if this continuous addi- 
tion to the amount of rotation were purely arbitrary and could have no 
counterpart in nature. But real bodies cool and contract in cooling, and, 
' M. Liapounoff contends that the ‘ pear-shaped’ figure is always unstable (‘ Sur 
un probléme de Tchebychef,’ Acad. Imp. des Sciences de St-Pétersbourg, 1905), but 
I cannot agree with this view—at least for the present. 
? See a paper by myself on ‘ Roche’s Ellipsoids and on Allied Problems’ com- 
municated to the Royal Society, January 1906. 
