546 The Genesis of Double Stars 



have started with A + a, and have watched the characteristic a 

 dwindling- to zero. When it vanishes we have reached a type which 

 may be specified as A ; beyond this point the type would be A - a or 

 would be impossible. 



Following the A + b type in the same way, b is at first well marked, 

 it dwindles to zero, and finally may become negative. Hence in short- 

 hand this second family may be described as A + b,. . . A,. . . A b. 



In each family there is one single member which is indistinguish- 

 able from a member of the other family ; it is called by Poincare a 

 form of bifurcation. It is this conception of a form of bifurcation 

 which forms the important consideration in problems dealing with the 

 forms of liquid or gaseous bodies in rotation. 



But to return to the general question, thus far the stability of 

 these families has not been considered, and it is the stability which 

 renders this way of looking at the matter so valuable. It may be 

 proved that if before the point of bifurcation the type A + a was 

 stable, then A + b must have been unstable. Further as a and b each 

 diminish A + a becomes less pronouncedly stable, and A +b less 

 unstable. On reaching the point of bifurcation A + a has just ceased 

 to be stable, or what amounts to the same thing is just becoming 

 unstable, and the converse is true of the A + b family. After passing 

 the point of bifurcation A + a has become definitely unstable and 

 A + b has become stable. Hence the point of bifurcation is also a 

 point of " exchange of stabilities between the two types 1 ." 



In nature it is of course only the stable types of motion which can 

 persist for more than a short time. Thus the task of the physical 

 evolutionist is to determine the forms of bifurcation, at which he 

 must, as it were, change carriages in the evolutionary journey so as 

 always to follow the stable route. He must besides be able to 

 indicate some natural process which shall correspond in effect to the 

 ideal arrangement of the several types of motion in families with 

 gradually changing specific differences. Although, as we shall see 

 hereafter, it may frequently or even generally be impossible to specify 

 with exactness the forms of bifurcation in the process of evolution, 

 yet the conception is one of fundamental importance. 



The ideas involved in this sketch are no doubt somewhat recondite, 

 but I hope to render them clearer to the non-mathematical reader by 



1 In order not to complicate unnecessarily this explanation of a general principle I have 

 not stated fully all the cases that may occur. Thus : firstly, after bifurcation A + a may 

 be an impossible type and A + a will then stop at this point ; or secondly, A + b may 

 have been an impossible type before bifurcation, and will only begin to be a real one 

 after it; or thirdly, both A + a and A + b may be impossible after the point of bifurcation, 

 in which case they coalesce and disappear. This last case shows that types arise and 

 disappear in pairs, and that on appearance or before disappearance one must be stable 

 and the other unstable. 



