Forces derivable front a Potential of the Second Degree. 57 



Application of Method of Mean Values. 



13. Let ti, ZT4a/3, ., t 3 be the mean lengths of material lines 

 parallel to the principal axes 2a, 2b, 2c, and let v be the volume of 

 the ellipsoid. Then it is easily proved from the results in my paper* 

 on the mean values of strains, &c., that the elastic increments in the 

 general case are given by 



ct.lt, =i(^/E){Pa 2 -^(Q6 z + Rc 2 )} ........ (70), 



.......... (71), 



(72). 



In (71) k denotes the "bulk modulus." 



For the gravitating nearly spherical ellipsoid, (70) gives 



Comparing (73) and (19) we see that in a gravitating perfect 

 sphere 



M,/f, a/a, 



or the reductions per unit length of a diameter and of the mean 

 parallel chord are identical. 



For the increment in volume in the gravitating nearly spherical 

 ellipsoid, (71) gives 



For the gravitating very flat spheroid of 7, we find 

 Stjti = Tr*fif* 2 ac(l >/)/5E, I 



.......... (75). 



c(l 2 i/)/5EJ 

 For the general case of an ellipsoid rotating about 2 a, we have 



iv/v - wV(&* + 



* ' Camb. Phil Soc. Trans.,' vol. 15, pp. 313337. 



t Agrees with formula (105), p. 335, ' Carab. Trans.,' loc. pit. 



r 2 



