ISO Prof. a. H. Darwin. Stability of the Pear-shaped [June 19, 



of order e 2 , and thence to find the kinetic energy T. The function E 

 is then equal to W + T. 



In order to attain the requisite degree of accuracy it is convenient 

 to regard the pear as being built up in an artificial manner. 



I construct an ellipsoid similar to and concentric with the critical 

 Jacobian, and therefore itself possessing the same character. The size 

 of the new ellipsoid, which I call J, is undefined • and is subject only 

 to the condition that it shall be large enough to enclose the whole 

 pear. The region between J and the pear being called E, I suppose 

 the pear to consist of positive density throughout J and negative 

 density throughout E. 



The lost energy of the pear consists of that of J with itself, say J J J ; 

 of J with E, which is filled with negative density, say - JE ; and of 



- E with itself, say JEE. This last contribution (which had baffled 

 me) must be broken into several parts. 



If we imagine J to be intersected by a family of orthogonal curves, 

 and if we suppose for the moment that the region E is filled with 

 positive matter, we may further imagine the matter lying inside any 

 orthogonal tube to be transported along the tube, and deposited on the 

 surface of J in the form of a concentration of positive surface density 

 + C. 



In the actual system E is filled with negative density, and we may 

 clearly add to this two equal and opposite surface densities + C and 



- C on J. The matter lying in the region E may then be regarded 

 as consisting of negative surface density - C, together with a double 

 system, namely negative volume density - E, conjoined with equal 

 and opposite surface density +C. This double system, say D, is 

 therefore C - E. 



The lost energy JEE may now be considered as consisting of three 

 parts, first, the energy of - C with itself, say JCC ; secondly, that of 

 D with itself, say JDD ; and thirdly of - C with D. This third item 

 is obviously equal to - CC + CE, and therefore JEE is equal to 



- JCC + CE + -i-DD. Thus W, the gravitational lost energy of the 

 pear, may be written symbolically — 



\ r J J - JE + CE - WC + JDD. 



In this discussion no attention has as yet been paid to the rotation, 

 but fortunately it happens that the introduction of this consideration 

 actually simplifies the problem, for if we suppose |JJ and JE to mean 

 the lost energies of J with itself and with E on the supposition that 

 the mass is rotating with the angular velocity of the critical Jacobian, 

 the formulae become much more tractable than would otherwise have 

 been the case. 



The inclusion of part of the angular velocity in this part of the 

 function only leaves outstanding the excess of the kinetic energy of 



