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alluded to above in a footnote； the speed diminishes。）； the equator tends to bulge outwards at two diametrically opposite points and to be flattened midway between these protuberances。 The specific difference in the new family； denoted in the general sketch by b； is this ellipticity of the equator。 If we had traced the planetary figures with circular equators beyond this stage A； we should have found them to have become unstable； and the stability has been shunted off along the A ＋ b family of forms with elliptic equators。
This new series of figures； generally named after the great mathematician Jacobi； is at first only just stable； but as the density increases the stability increases； reaches a maximum and then declines。 As this goes on the equator of these Jacobian figures becomes more and more elliptic； so that the shape is considerably elongated in a direction at right angles to the axis of rotation。
At length when the longest axis of the three has become about three times as long as the shortest （The three axes of the ellipsoid are then proportional to 1000； 432； 343。）； the stability of this family of figures vanishes； and we have reached a new form of bifurcation and must look for a new type of figure along which the stable development will presumably extend。 Two sections of this critical Jacobian figure； which is a figure of bifurcation； are shown by the dotted lines in a figure titled 〃The 'pear…shaped figure' and the Jocobian figure from which it is derived〃 （Fig。 3。） comprising two figures； one above the other： the upper figure is the equatorial section at right angles to the axis of rotation； the lower figure is a section through the axis。
Now Poincare has proved that the new type of figure is to be derived from the figure of bifurcation by causing one of the ends to be prolonged into a snout and by bluntening the other end。 The snout forms a sort of stalk； and between the stalk and the axis of rotation the surface is somewhat flattened。 These are the characteristics of a pear； and the figure has therefore been called the 〃pear…shaped figure of equilibrium。〃 The firm line shows this new type of figure； whilst； as already explained； the dotted line shows the form of bifurcation from which it is derived。 The specific mark of this new family is the protrusion of the stalk together with the other corresponding smaller differences。 If we denote this difference by c； while A ＋ b denotes the Jacobian figure of bifurcation from which it is derived； the new family may be called A ＋ b ＋ c； and c is zero initially。 According to my calculations this series of figures is stable （M。 Liapounoff contends that for constant density the new series of figures； which M。 Poincare discovered； has less rotational momentum than that of the figure of bifurcation。 If he is correct； the figure of bifurcation is a limit of stable figures； and none can exist with stability for greater rotational momentum。 My own work seems to indicate 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。）； but I do not know at what stage of its development it becomes 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 has 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。
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 consequence 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's figure for an advanced stage of development is shown in a figure titled 〃Section of a rotating cylinder of liquid〃 （Fig。 4。）； but his calculations do not enable him actually to draw the state of affairs after the rupture of the neck。
There are certain difficulties in admitting the exact parallelism between this problem and ours； and thus the final development of our pear…shaped figure and the end of its stability in a form of bifurcation remain hidden from our view； but the successive changes as far as they have been definitely traced are very suggestive in the study of stellar evolution。
Attempts have been made to attack this problem from the other end。 If we begin with a liquid satellite revolving about a liquid planet and proceed backwards in time； we must make the two masses expand so that their density will be diminished。 Various figures have been drawn exhibiting the shapes of two masses until their surfaces approach close to one another and even until they just coalesce； but the discussion of their stability is not easy。 At present it would seem to be impossible to reach coalescence by any series of stable transformations； and if this is so Professor Jeans's investigation has ceased to be truly analogous to our problem at some undetermined stage。 However this may be this line of research throws an instructive light on what we may expect to find in the evolution of real stellar systems。
In the second part of this paper I shall point out the bearing which this investigation of the evolution of an ideal liquid star may have on the genesis of double stars。
II。
There are in the heavens many stars which shine with a variable brilliancy。 Amongst these there is a class which exhibits special peculiarities； the members of this class are generally known as Algol Variables； because the variability of the star Beta Persei or Algol was the first of such cases to attract the attention of astronomers； and because it is perhaps still the most remarkable of the whole class。 But the circumstances which led to this discovery were so extraordinary that it seems worth while to pause a moment before entering on the subject。
John Goodricke； a deaf…mute； was bor