Forces derivable from a Potential of the Second Degree. 55 , 



_ - 



' (1-^) (3a 4 + 2a 2 6 2 +36 4 ) 



.. (58). 



2/y and y mny be got from aa; and zx respectively by interchanging 



x with y and a with 6. To a first approximation yz and zx are 

 negligible. 



Neglecting z 2 /c 2 in these formulae, we obtain results applicable to 

 the central portion of the long ellipsoid ; these results are identical 

 with those I have previously obtained for a long elliptic cylinder,* 

 the axes of whose elliptic section are 2a and 2fc, rotating about the 

 cylindrical axis. To deduce results for the elongated ellipsoid from 

 those found for the infinite cylinder, we write 1 z 2 /c 2 for 1 in the 

 constant terms in $ x , s y> s z , and A, in the coefficient of x in and in 

 that of y in ft and we multiply the expression for 7 by 1 ^z~jc^. 



The strain and displacement parallel to the long axis are of special 

 interest ; they are 



= - 



(59), 



Denoting by 2 1 the length of the long elliptic cylinder, we have 



(61). 



This enables the values of fic/c, for y = 0'25 and 6/a = 0, - 2, 0'4, 

 0'6, 0*8, and 1, to be written down from Table XV, p. 159, of my 

 paper on the elliptic cylinder. Table XVI of that paper gives values 

 of caja and Sb/b which apply unchanged to the long ellipsoid; while 

 Tables XIII and XIV give its limiting angular velocity on the stress- 

 difference and greatest strain theories. 



The formula 



.. (62) 



shows that of the principal transverse axes the longer is that which, 

 even proportionately, is most extended. 



* 4 Phil. Mag.,' Aug., 1892, formulae (80) to (S3), p. 156. 

 VOL. LVIJI. 



