42 



Mr. I. Todbunter on Jacobi's Theorem. 



On Jacobi's Theorem respecting the relative eqnilibrinm of a 

 Revolving Ellipsoid of Fluid, and on Ivory's discussion of 

 the Theorem/' By L Todhunter, M.A., F.R.S., late Fellow 

 of St. John's College, Cambridge. Received November 23, 

 1869*. 



L The late James Ivory contributed to the Philosophical Transac- 

 tions various memoirs on the subject of the equilibrium of fluids and the 

 figure of the earth : the memoirs will be found in the volumes for 1824, 

 1831, 1834, and 1839. Ivory objected to the received theory of the equi- 

 librium of fluids, and advocated some peculiar opinions at great length, 

 and with much repetition. I do not propose now to criticise these memoirs ; 

 I will merely state that I consider them to be altogether unsatisfactory. 



2. There is, however, one theorem in the general subject to which I 

 now propose to draw attention, namely, Jacobi's theorem respecting the 

 possibility of the relative equilibrium of an ellipsoid of fluid having three 

 unequal axes and revolving about the least. Ivory discussed this theorem, 

 and his errors are so numerous and so singular, that I have thought it 

 would be desirable to place the corrections before the Society which origi- 

 nally received and published Ivory's communications. In correcting Ivory's 

 errors and supplying his defects, I shall add something to the discussions 

 which have hitherto been given of the theorem itself. It will be seen 

 as we proceed that one of Ivory's errors has been already noticed and 

 corrected. 



3. Ivory first alluded to the matter in the memoir of 1 834, which was read 

 to the Royal Society a few months before Jacobi announced his discovery 

 of the theorem. Of course at that date Ivory held the common opinion, 

 that the relative equilibrium of a revolving ellipsoid with three unequal 

 axes was impossible. But he does not merely acquiesce in the erroneous 

 opinion, he attempts to demonstrate it in the following manner : — 



" Further, the figure of the fluid in equilibrium can be no other than 

 a spheroid of revolution. Draw a plane through the axis of rotation and 

 any point (xyz) in the surface of the fluid. This plane will contain that 

 part of the attraction of the spheroid which is parallel to the axis of rota- 

 tion, or to the coordinate co : it will also contain the centrifugal force 

 directed at right angles from the axis of rotation. The same plane will also 

 contain the resultant of the attractions parallel to y and z ; for if it did 

 not, the resultant might be resolved into two forces, one contained in the 

 plane, and the other perpendicular to it ; and the force perpendicular to 

 the plane would partly act in a direction touching the surface of the 

 spheroid, which is inconsistent with the equilibrium of the fluid. Where- 

 fore, the whole attractive force at any point in the surface of the spheroid 

 is contained in a plane passing through the point and the axis of rotation ; 



* Bead Jan. 20, 1870. See vol. xviii. p. 171. 



