54 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND 



If we substitute these values in equation (92), we find 



= -0'00031998 + 0-00009111 + < 00022829 = -O'OOOOOOS. 



The fact that the error occurs only in the seventh place of decimals adequately 

 verifies DARWIN'S calculations, but the tendency of small errors to accumulate in 

 computation is forcibly illustrated by the circumstance that in the above equation 

 the final error is as much as one-six-hundredth part of the whole value of the middle 

 term. 



With the values just obtained for a, ft', y , I find 



y') n -- -G'00094878, 

 (So.' lAA + ^LvB + y'lAc) = -0-00094885, 



verifying that equation (79) is satisfied, again as far as the sixth place of decimals. 



24. With a view to subsequent computations, it is convenient to take a standard 

 set of values such that a' = 1. These values are found to be 



a ' = _l, fjf = G'3081810, y' = 0'1604294, 



and with these we have, by equation (65), 



3 a ' + 3' + ' = Q-506278. 



These numerical values substituted in equation (64) will give the equation of 

 POINCARK'S pear-shaped figure as far as small terms of the first order. 



The Pear-shaped Figure Calculated to the Second Order. 



25. The question as to whether the pear-shaped figure is stable depends upon the 

 change effected by the distortion upon the angular momentum of the ellipsoid. But 

 (cf. 18) the first-order distortion so far considered can be easily seen to produce no 

 effect at all upon the angular momentum of the figure. It is therefore necessary to 

 proceed to terms of a higher order, and we now consider terms of the second order. 



The first-order terms have been found to be given by 



yf + K ), ........ (93) 



with (cf. equation (47)) ?V= -iDtt^ The value of n 2 will be given by equation (48), 

 in which u r is to be assigned the value (93), and w will be taken to be given by 



to = L^ + M^ + Nr + 2/,, 2 f + 2m^ 2 + 2wfV + 2(p 2 + g^-Hrf 2 ) + s. . . (94) 



It has to be shown that this value for u> makes it possible for the figure distorted 

 to the second order in this way to be a figure of equilibrium. 



