REVIEWS 681 



The volume can be divided almost naturally into two parts. The first 

 part, extending to the end of Chapter VIII, contains the mathematical 

 framework on which, in the second part, from Chapter IX to the end of 

 the book, the author has attempted to fit the facts of observational astronomy. 

 The mathematician will probably be interested mainly in the first portion, 

 whereas the practical astronomer will be inclined to take for granted the 

 results derived therein and will concern himself with their application in the 

 second part, which is tolerably free from mathematics : this portion is, 

 naturally, of a somewhat speculative nature. 



The introductory chapter contains a general survey of the problem, 

 commencing with a description of the solar system and of the phenomena 

 of binary stars, spiral and other nebulae, and of star-clusters. Any com- 

 plete theory of cosmogony must account for these five different types of 

 structure by discovering their origins and proving that they lead to the 

 observed structures. A brief critical survey of various earlier theories of 

 cosmogony then follows, none of these being found free from objections. 



The following six chapters contain the theoretical investigation into the 

 configurations of equilibrium of rotating masses, together with a discussion 

 of their stability. The case of incompressible masses is first treated, fol- 

 lowed by that of compressible and non-homogeneous bodies. The dyna- 

 mical principles upon which the investigation is based, amongst which 

 Poincare's conception of series of continuous configurations of equilibrium 

 plays an important role, are lucidly explained in the second chapter. The 

 mathematical analysis involved in this investigation is of great complexity, 

 and, though elegant in principle, very cumbrous expressions have to be 

 dealt with. Each step in the argument is given, though for some of the 

 more lengthy reductions the reader is referred to the original papers. In- 

 cluded in these chapters is a discussion of the stability of the famous pear- 

 shaped figures, and it is shown that, although Darwin and Liapounoff obtained 

 contradictory results, yet both were in error, in so far as they proceeded 

 only to the second order of small quantities, whilst Jeans shows that to this 

 order the stability is indeterminate, so that it becomes necessary to con- 

 sider third order terms. The complexity of the discussion is thereby very 

 considerably increased, but Jeans has carried it through successfully and 

 has shown that the figures are unstable. 



The labour of computation involved in tracing out the sequence of con- 

 figurations of the pear-shaped series after instability has set in is so heavy as 

 to be prohibitive. Jeans has, however, discussed the analogous problem 

 for the two-dimensional case, and the similarity between the two problems 

 is so close before instability ensues that it is very probable that it persists 

 afterwards. For the two-dimensional case, the pear-shaped figure is found 

 to break up into two separate masses in mutual rotation. 



In addition to the rotational problem, Jeans has also discussed what he 

 calls the tidal problem, i.e. the motion of a primary mass as tides are raised 

 in it by the continued approach and ultimate recession of a secondary mass, 

 and also the double-star problem, i.e. the motion of two stars revolving round 

 one another, a secular change being supposed to occur in their distance 

 apart. In all three problems, the only figures of stable equilibrium are ellip- 

 soids and spheroids. The dynamical motion in the two latter problems is 

 also shown, with some degree of certainty, to lead to fission into detached 

 masses ; in the tidal problem any finite number of detached masses may 

 result ; in the double-star problem, a large number of small masses are 

 formed. 



The discussion is then extended to the much more complex case of com- 

 pressible and non-homogeneous masses ; it is shown that in each of the 

 three different problems there are these two possible methods of break-up, 

 one by fission, as in the case of incompressible mass, and the other, in which 



