58 REPORT— 1882. 



denoted by V, then the combination of Meyer's investigation and Laplace's, 

 gives the following results. If an ellipsoid is to be a form of equi- 

 librium, V must lie between the values V = and V = -2246. ... If V 

 lies between and Vq = •18711 . . . then for a given value of V there is 

 one ellipsoidal form with unequal axes, and two spheroidal forms, whilst 

 for V = Vq the former coalesces into that spheroidal form which has the 

 less axis of rotation. When V is between Vq and V there can only be hvo 

 spheroidal forms, which for V = V coalesce into one. The ultimate 

 spheroid to which the ellipsoid approximates when V := Vq has the ratio 

 of its axes equal to •5827. . . . For V = the limit for the ellipsoid is the 

 circular cylinder, whilst the spheroids are — one a sphere and the other an 

 infinite disc. It is clear that the most natural datum to take is not the 

 angular velocity but the angular momentum, which remains constant, 

 however the fluid may change its form. This was a point of view adopted 

 by Laplace in treating of the spheroidal form, and Liouville ' took up 

 Meyer's problem in the same way in a paper read before the French Aca- 

 demy of Sciences in 184.3, in which he showed that the ellipsoidal form 

 with unequal axes is only possible provided the ratio of the angular mo- 

 mentum to the mass is greater than a certain limit, thus differing from 

 the spheroids, which are forms of equilibrium for any given angular mo- 

 mentum. What happens when the angular velocity of a spheroid is too 

 great for it to keep its form ? This could be answered generally from 

 the foregoing theories, viz., that tbe spheroid would become flatter, so 

 decreasing its angular velocity, and that it would vibrate about some 

 mean position ; but whether its external form would always be spheroidal, 

 or what the precise manner of the movement might be, could not be 

 decided. This question was answered, and the complete theory of the 

 motion of sjiheroids of fluid investigated, in a posthumous paper by 

 Dirichlet,2 edited and enlarged by Dedekind. The extremely beautiful, 

 and in its fundamental idea simple, theory of Dirichlet threw open to 

 mathematicians a new and rich field for further investigations, of 

 which they were not slow to avail themselves, so that now it may be said 

 that we know the general pi'operties of the motion of a mass of fluid 

 moving with a free ellipsoidal surface under its own attraction. Dirichlet's 

 first conception dates from the winter of 1850-1857, so his editor, Dedekind, 

 says ; but the author, wishing to extend them further, did not publish his 

 results in full, and they did not appear until 1859, after his death, when 

 Dedekind published them with some further results of his own. I will 

 first attempt to give a genei'al idea of his method, then refer to the chief 

 results of his investigation, and afterwards pass on to notice the work 

 done by other mathematicians, following on the lines laid down by him. 



Considering that the Lagrangrian method of treating fluid motion is 

 better fitted than the Eulerian when the boundary surface changes with 

 the time, he asks the question. Is it possible to have the co-ordinates of a 

 particle at any time linear functions of its original co-ordinates, and if 

 BO, to what kind of motion does it refer ? It is clear at once that those 

 particles originally lying on an ellipsoid must always do so, though not 



' This was piiblished in the Additions a la Connaiasance ties Temps for 1846, and 

 also in 1851 in lAonrille's Journal, xvi. p. 241, under the title ' Sur les figures ellip- 

 so'idales i\ trois axes inegaux, qui peiivent convcnir ;\ I'equilibre d'un masse liquide 

 homogene, douee d'un movivemeut de rotation.' 



- ' Ueber ein Problem dor Hydrodj'namik,' AbJiand, ¥6n. Gcs. Wiss. Gott. viii. p. 

 1, and Borck. Iviii. p. 181, 



