[ 40 ] 



VI. On DedekincFs Theorem concerning the Motion of a 

 Liquid Ellipsoid under its own Attraction. By A. E. H. 

 Love, Fellow of St. Johns College, Cambridge** 



TT1HE interesting problem, to determine the general motion 

 JL of a mass of liquid in the form of an ellipsoid which 

 moves, subject to its own attraction, in such a way as to remain 

 ellipsoidal and of constant volume, was the subject of a posthu- 

 mous paper by Dirichlet, edited, with additions, by Dedekind, 

 in Borchardt's Journal, vol. lviii. (1859). 



Dirichlet set out from the assumption that the coordinates 

 of any liquid particle are at any instant linear functions of 

 their initial values, and showed that all the conditions could 

 be satisfied. The investigations of Greenhillf, completed by 

 Basset J, lead to the conclusion that the most general type of 

 motion within the ellipsoid can be thus expressed. 



Dedekind's most important contribution to the subject is a 

 theorem of reciprocity, according to which there corresponds 

 to any given motion of the ellipsoid a certain other correlated 

 motion. It is the object of this Note to give a proof of this 

 theorem, and especially to discuss the motion which is thus 

 correlated with the well-known Jacobian form of the rotating 

 ellipsoid. This case of motion is usually referred to as 

 "Dedekind's ellipsoid;" and the physical description of it is 

 that the ellipsoidal form remains fixed in space, while the 

 liquid moves about inside it. Regarded from a tidal point of 

 view, we might say that it corresponds to a case of a fixed 

 finite tide with no disturbing body. 



Dedekind's Theorem. 

 Let a, b, c be the axes of the ellipsoid at any time t, and 

 x, y, z the coordinates of a fluid particle at this time referred 

 to axes fixed in space whose origin is at the centre of the 

 ellipsoid ; a , b , c , x , y , z the initial values of these quan- 

 tities at time t = 0, and let us suppose that at this time the 

 axes of coordinates coincide with the axes of the ellipsoid. 

 Then we take 



(1) 





°Q 



+z 



3" 



A' 



4-7/ 7J ° 

 °0 



+ z 2 



z 



- ■> 



Co 



a 



4-7/ - /o 

 °0 



+ ^3 



*0 



— 



Co 



J 



* Communicated by the Author. 



t Proc. Camh. Phil Soc. vols. iii. & iv. (1880 and 1882). 



X Proc. Lond. Math. Soc. vol. xvii. (1880). 



