AND THE NEBULAR HYPOTHESIS. 457 



This gives A, for the general tide raised by the field v n S m . The condition for a 



point of bifurcation is ?> = 0, or 



....... - ..... < 38 > 



Thus the points of bifurcation, if any, are still determined by the intersections of 

 the graphs in fig. 1, except that the graph of , must be supposed decreased vertically 



o 



in the ratio 1 -- to 1 . 



27T<7 



We may, if we please, imagine that we start with very small rotation, and allow the 

 rotation progressively to increase, this increase being accompanied in imagination by a 

 greater and greater flattening of the graph of u,. 



It is clear that under all circumstances the curve which will first be intersected by 

 the flattened graph of ?<, will be the graph of u 3 . It is further clear that the 

 requisite value of iv* is least when *a = 0, and progressively increases as *a increases, . 

 at any rate up to KO, = TT. 



This means that in the first place the circular vibration will invariably become 

 unstable through a vibration proportional to a second harmonic, so that the first point 

 of bifurcation reached will be one such that the spheroidal form gives place to an 

 ellipsoidal form. If the rotation is so small that the problem may be treated as a 

 statical one, there will be no question as to there being an actual exchange of 

 stabilities at the point of bifurcation, for clearly v n changes sign at this point. Thus 

 for rotation greater than that at the point of bifurcation, the spheroidal form will be 

 definitely unstable, and the ellipsoidal form definitely stable, at least until the next 

 point of bifurcation is reached. 



Our result shows, in the second place, that the masses which become ellipsoidal for 

 the smallest values of it? are those for which *a is smallest. To put it briefly, the 

 mass which is most unstable when it begins to rotate is the incompressible mass a 

 somewhat unexpected result. 



For any value of *, the value which iff 1 must have for the spheroidal form to 

 become unstable is (cf. equation (38)) 



and when K a = 0, the value of u-Ju^ = I (cf. 4). 



Thus our equations would make the spheroidal mass of incompressible fluid first 



become unstable when - = "400, but these equations have only been obtained on 



Zww 



the supposition that is so small that its squares may be neglected, a supposition 



2x<r 



which is now seen d posteriori to be hardly admissible. Probably the result 

 obtained are qualitatively true, but quantitatively unreliable. In point of fact the 



3 O 2 



