18 Introductory Chapter [OH. i 



may be equally expressed in the form 



(0 = 1,2,3,...) 



which may be taken to represent the distances of nuclei along the arms of 

 an equiangular spiral. 



In a somewhat different class come hypotheses, such as those of Faye* 

 and Ligondesf, which try to prove that our system originated out of a 

 swarm of meteorites in which order has been produced out of disorder by 

 collisions, in opposition to the laws of statistical mechanics j. 



Our task in the present essay is not to discuss these and other theories in 

 detail ; it is rather to obtain mathematical evidence bearing on the general 

 problem of evolution, incidentally perhaps examining to what extent the 

 speculative theories which have just been described are tenable. Many of 

 these theories, however, have already been condemned by the recent advances 

 in observational astronomy. For in many cases the theories were not based 

 on abstract knowledge of the properties of matter or on dynamical laws ; 

 they rather exhibited a tendency to be based on the latest observational 

 knowledge with which their authors were acquainted. Up to the discovery 

 of the spiral nebulae, most theories of cosmogony tried to prove that Saturn's 

 rings (the most sensational astronomical objects then known) formed an 

 intermediate stage in the evolution of planetary systems : since the discovery 

 of spiral nebulae, the tendency has been to try to prove that the spiral 

 nebulae form the link in question. The more scientific method of procedure 

 is to limit the investigation to the abstract problem of the behaviour of 

 masses of astronomical matter under varying dynamical forces; when the 

 solution of this problem has been carried to the limit of our mathematical 

 resources, we shall be in a position to survey the different types of formation 

 that may be expected to be evolved, and possibly not much speculation will 

 be required to identify them with observed forms. Thus the immediate 

 object of the present essay will be to collect and arrange the results of the 

 various researches which have resulted in progress towards the solution of 

 this abstract problem, adding to them and amplifying them wherever we can. 



The dynamical forces which can act on astronomical matter are its own 

 gravitation, which must always be taken into account: the -gravitational 

 forces from other bodies, which we may for brevity describe as tidal forces ; 

 the forces arising from rotation; the forces arising from collisions, impacts, 

 bombardments, etc. Our problem is to find out as much as we can about 

 the behaviour of matter under such forces, paying attention especially to 

 effects of a secular or evolutionary nature. 



* Sur VOrigine du Monde. (Paris, Gauthier-Villars, 1884.) 



t formation M&canique du Sysieme du Monde. (Paris, Gauthier-Villars, 1897.) 



Cf. Poiucare, Lemons sur les Hypotteaes Cosmogoniques, Chapters IV and V. 



