NATURE 



[March ii, 1920 



of a finite mass of rotating incompressible fluid. 

 Year after year, questions of the multiplicity of 

 possible figures of equilibrium have been almost 

 incessantly before us, and yet it is only now, under 

 the compulsion of finishing this second edition of 

 the second part of our first volume, with hope for 

 a second volume abandoned, that we have suc- 

 ceeded in finding anything approaching to full 

 light on the subject." 



The full light, it must be admitted, was rather 

 dim, especially as such results as had been 

 obtained were published without proof. But it 

 sufficed to lead Poincar6 to write a celebrated 

 memoir on the subject, and this, with the contem- 

 porary and independent work of Liapounoff, has 

 been the germ of all subsequent advance. 



The first step had been taken by Maclaurin, 

 who showed that the spheroid was a possible 

 figure of equilibrium. The second solution was 

 found by Jacob! in the form of an ellipsoid with 

 three unequal axes. In the cosmical problem the 

 whole mass and the moment of momentum are 

 given 'constants; the angular velocity increases 

 as the state of contraction advances, but so slowly 

 that the development follows a succession of equi- 

 librium figures. Thus a body traces out the series 

 of Maclaurin to the point where it meets the series 

 of Jacobi, and where secular stability is inter- 

 changed between the two series. Proceeding along 

 the second series, it comes to the first point, dis- 

 covered by Poincare, where another possible series 

 intersects. Here the Jacobian series becomes 

 unstable, and it was a question whether the 

 stability passed over to the series of' deformations, 

 or whether it disappeared completely at this point, 

 in which case the figures of statical equilibrium 

 would come to an abrupt end and be followed by 

 a rapid change under dynamical conditions. 



It was not in Poincare's nature to embark on the 

 complicated arithmetic needed to solve the ques- 

 tion ; but this part of the work was supplied by 

 Liapounoff and by Darwin, who arrived at oppo- 

 site conclusions, the latter maintaining that the 

 deformed figure was stable. These three writers 

 all used Lamp's functions in the discussion, and 

 carried the development to the second order. One 

 cannot help feeling that, in spite of his courage, 

 Darwin was in this instance trying to stretch a 

 bow a little beyond his strength. At any rate, the 

 important problem remained undecided for some 

 . years. Mr. Jeans began his attack on it by forg- 

 ing a lighter and handier weapon, described in 

 chap. iv. of the present work, on the gravitational 

 potential of a distorted ellipsoid. His next step 

 was to show that no conclusion could legitimately 

 be drawn from a development to the second order ; 

 NO. 2628, VOL. 105] 



and finally, on proceeding to the third order, he 

 proved definitely that the figure at the point where 

 the series bifurcate is unstable, thus closing a 

 dispute remarkable in the case of a definite issue 

 between authorities so eminently qualified. After 

 this signal achievement as regards the incom- 

 pressible fluid mass Mr. Jeans extended his 

 researches to rotating masses of compressible 

 and heterogeneous fluid, hitherto an almost 

 untouched field. In following out the develop- 

 ment of such bodies as exemplified in different 

 selected models, he has shown always the same 

 originality, resource and power. 



In the present essay, which will be warmly wel- 

 comed, Mr. Jeans brings together these and other 

 related researches in a connected form, but at the 

 same time he adds so much of the work of his 

 predecessors that his own is seen in its proper 

 setting, and the whole book forms a fairly com- 

 plete treatise on the subject. The earlier chapters 

 provide that firm mathematical foundation to 

 which the author has contributed so largely, while 

 the later chapters deal in turn with the different 

 classes of celestial objects to which the theory can 

 be applied — rotating nebulae, star clusters, binary 

 and multiple stars. The origin of the solar system, 

 the very point at which speculations of this order 

 began, remains apparently more elusive than ever. 

 The later part of the book can be read with profit 

 by many to whom the power of appreciating the 

 earlier mathematical chapters has been denied. It 

 will be found exceedingly interesting, and wilt 

 repay the most careful attention. Here the specu- 

 lative element necessarily enters, and the per- 

 manent value which belongs to the abstract 

 problems definitely solved cannot be assumed. 

 But ingenuity and a wide knowledge are always 

 in evidence, and the essay should have an imme- 

 diate value equally in limiting the area of profit- 

 able speculation and in suggesting lines which 

 can be controlled by observation. 



Of the technical excellence of the production, 

 which is always a point of real importance in a 

 mathematical text, it is unnecessary to say more 

 than that it is worthy of the Cambridge University 

 Press. There is an obvious, and therefore harm- 

 less, misprint in equation (72) (p. 38), and 

 " Meyers " (p. 248) for " Myers " betrays an un- 

 verified quotation. On p. 2, "parallaxes are less " 

 should read "greater." But these are trifling 

 exceptions to the rule of accuracy. Beautiful pic- 

 tures like the photographs of selected nebulae in- 

 cluded by Mr. Jeans are an unusual feature in a 

 mathematical work. They have been supplied from 

 the Mount Wilson Observatory, and are master- 

 pieces of their kind. H. C. P. 



