482 MR. J- H. JEANS ON GRAVITATIONAL INSTABILITY 





the crust to be </, whence it follows that the mean density of the whole mass will be 

 roughly equal to a density defined by 



and the value of w^flirQ will be approximately the same as the quantity w 2 f'2irp which is 

 computed from observations of binary stars. Values of w 2 /2v8 are given in the last 

 column of the table on the preceding page ; the value 0'40 corresponding to tc'a = 

 (no core) being inserted from the result of the previous analysis ( 13). As before, 

 the numbers are not numerically accurate, but their general trend may be expected to 

 reveal the general trend of the true series of numbers. It at once appears that the 

 values of w^fe-rd are surprisingly steady : there is certainly no rapid decrease in their 

 amount as // increases. 



Summary and Conclusion. 



25. The problem we have had under consideration has been that of testing whether 

 the behaviour of a rotating mass of compressible heterogeneous matter differs very 

 widely from that of the incompressible homogeneous mass which has been studied by 

 MACLAUEIN, JACOBI, POINCARE, and DARWIN. The result obtained can be summed 

 up very briefly by saying that the ideal incompressible mass has been found to supply 

 a surprisingly good model by which to study the behaviour of the more complicated 

 systems found in astronomy. The problem especially under consideration has been 

 that of determining the amount of rotation at which configurations of revolution 

 (e.g., spheroids) first become unstable. In so far as we have been able to examine the 

 question, it appears probable that the compressible mass will behave, at least up to this 

 point, in a manner almost exactly similar to the simpler incompressible mass, and 

 results obtained for the latter will be nearly true, both qualitatively and quantitatively, 

 for the former. The compressible mass, set into rotation, will apparently pass through 

 a series of flattened configurations very similar to the Maclaurin spheroids ; it will 

 then, for just about the same amount of rotation (as measured by w 2 jp), leave the 

 symmetrical form and assume a form similar to the Jacobian ellipsoids. Beyond this 

 stage our analysis has not been able to deal with the problem. Indeed, strictly 

 speaking, our analysis has hardly been able to carry this far. A question of 

 importance has been whether the quasi-spheroidal form for a compressible mass does 

 not become unstable for a much smaller value of iv* than the incompressible mass, and 

 whether the instability does not set in in a different way. These questions we have 

 been able to answer, with, I think, a very high degree of probability, in the negative. 

 The whole matter is of necessity one of probability only, and not of certainty, for the 

 general heterogeneous compressible mass is not amenable to analysis until a great 

 number of simplifying assumptions have been made. 



It was first pointed out that a compressible mass has an infinitely greater number 



