Mr. I. Todhunter on Jacobi's Theorem. 



43 



which obviously excludes ellipsoids with three unequal axes, and limits the 

 figures of equilibrium to spheroids formed by the revolution of an ellipsis 

 about the axis of rotation ; . . 



The error here begins with the sentence which I have put in italics ; the 

 resultant of the attractions parallel to y and z need not act in the plane 

 which Ivory specifies : the component which he obtains in a plane touching 

 the surface may be balanced by a like component arising from the attrac- 

 tion parallel to x and the the so-called centrifugal force. 



4. To the Philosophical Transactions for 1838 Ivory contributed a 

 memoir of ten pages on Jacobi's theorem. Ivory devotes a few sentences 

 to the history of the matter. He records the fact that Lagrange had 

 inferred that the figure of relative equilibrium must be a figure of revolu- 

 tion. He makes no allusion, however, to his own erroneous demonstration 

 in the volume for 1834. 



5. The object of the memoir seems to be twofold — to establish Jacobi's 

 theorem, and to deduce numerical results relating to the extreme possible 

 cases analogous to those which had long been known relating to the extreme 

 possible cases for an ellipsoid of revolution. The first object is attained ; 

 Jacobi's theorem is demonstrated in a manner resembling that which had 

 previously been used by Liouville. The second object Ivory fails to attain, 

 owing to an error in his process. 



6. In the second page of the memoir there is an error in mechanics re- 

 sembling that which we have already noticed in Art. 3. At any point in 

 the surface of an ellipsoid, let the normal to the surface be drawn ; and 

 let it be terminated by the principal plane which is perpendicular to the 

 axis of rotation : let p be the length of this straight line. At the same 

 point in the surface draw a straight line in the direction of the resultant 

 of the attraction of the whole mass of the ellipsoid, and let it be terminated 

 by the same principal plane ; let p 1 be the length of this straight line, then 

 Ivory says : — 



" Let a denote the third side of the triangle which has p and p f for its 

 other sides : then a will represent the only force which, together with the 

 attractive force jo', will produce a resultant in the direction of p at right 

 angles to the surface of the ellipsoid." 



This statement is quite wrong. Any straight line which is in the same 

 plane as the normal at any point, and the direction of the resultant attrac- 

 tion at that point, may be taken for the direction of such an additional force 

 as Ivory requires ; and the magnitude of the force can then be properly 

 determined. 



7. In order to render the discussion of Ivory's memoir readily intel- 

 ligible, it will be necessary to indicate briefly the demonstration of Jacobi's 

 theorem. 



Let the equation to an ellipsoid be 



