78 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, ETC. 



The equations necessary for equilibrium are 



) = 0, ifaEa+JE.+JE.) = 0, (/ = /;, & c .), (172) 



oe 

 while the equations provided by DARWIN'S methods would be 



= 0, i(,5E 6 ) = 0, 



and it will readily be seen that there are more equations than can be satisfied by the 

 variables at our disposal. The conclusions of this section are put forward with the 

 utmost diffidence, but to the present writer they seem inevitable. 



40. Assuming that no glaring error has been made, the present investigation seems 

 to indicate that so far the knowledge we have as to the stability of the pear-shaped 

 figure amounts to absolutely nothing. The required knowledge can only be obtained 

 by carrying the figure of the pear to a still higher approximation. In the parallel 

 investigation on cylinders it was found that the stability could be examined as soon 

 as the figure was determined as far as third-order terms, and doubtless the same will 

 prove to be the case in the present problem. Fortunately the method of the present 

 paper is one which lends itself to indefinite extension, limited only by labour of 

 computation, so that it ought to be possible to proceed to third-order terms and 

 determine the stability of the pear, and if the pear then proves to be stable, to pro- 

 ceed to higher orders and so examine the series of pear-shaped forms. In the previous 

 investigation on cylindrical figures it was found that an expansion as far as fifth-order 

 terms gave a good approximation to the pear-shaped figure up to a stage where it 

 was obviously just about to divide into two separate masses. It is only in the hope 

 that I shall be able to carry the present investigation further, that I have ventured 

 to put forward the present somewhat lengthy piece of work which has so far led 

 only to such negative and disappointing conclusions. 



