126 Cataclysmic Motion [CH. vi 



\ 

 and R its distance. The critical value of R, as we have already seen 



(equation (86)), is 



R = 21984 f T r (378). 



\MJ 



When fju changes so rapidly that the forces may be regarded as impulsive; 

 the turning point is determined by equation (364), and it is found that this 

 will lie beyond the critical eccentricity '947741 if 



Left > 0-675/5* (379). 



126. In general we must contemplate not only cases in which the 

 spheroid runs over one point of bifurcation before reaching the turning point, 

 but" also cases in which it runs over several points of bifurcation. Some- 

 where near eccentricity e = "9477, a third harmonic displacement will become 

 unstable, and the spheroid will give place to a pear-shaped figure. A furrow 

 will develop near the middle plane of the spheroid and this will increase 

 rapidly (approximately exponentially) with the time. But meanwhile the 

 eccentricity of the spheroid may continue to increase, and it may be that 

 before the pear-shape is much developed, a second point of bifurcation will 

 be reached, namely that corresponding to a fourth harmonic displacement. 

 At this stage two new furrows begin to form, but these, like the former 

 pear-shaped furrow, will be forming in a spheroid which may simultaneously 

 be elongating itself with considerable velocity. When, or if, the next point 

 of bifurcation is reached, three more new furrows may begin to form, and 

 so on. 



Fig. 23 shews rough drawings (partly conjectural) of spheroids with the 

 furrows produced on passing the earlier points of bifurcation. Little doubt 

 will be felt that such figures will in time break up into a number of separate 

 detached pieces. 



127. So far we have been considering only an idealised mathematical 

 problem ; in nature there will be innumerable complications, and we must 

 try to calculate the effect of the more important of these. 



We have supposed the tidal potential to be M' (x 2 ^y*- %z z )IR\ which 

 is the potential either of a spherical mass, or of a mass of any shape at a 

 great distance. In an actual problem the potential will be more complicated 

 than this, for not only will the secondary mass not be spherical but the 

 shapes of the primary and secondary will influence one another, as in Darwin's 

 problem considered in 60. It is however not difficult to shew* that in 

 the most general case the motion is, in its main characteristics, entirely 

 similar to that just discussed. The numerical results are slightly altered, 



* The question is discussed in detail in a paper " The Motion of Tidally Distorted Masses," 

 R.A.S. Memoirs, Vol. 72. 



