1873.] On an Erroneous Extension of Jacobi's Theorem. 



119 



II. " Note on an Erroneous Extension of Jacobi's Theorem.'" By 

 Isaac Todhunter, M.A., F.R.S. Received October 25, 1872. 



1. It is well known that Jacobi discovered the possibility of the relative 

 equilibrium of a mass of homogeneous fluid which is in the form of an 

 ellipsoid and rotates with uniform angular velocity round the least prin- 

 cipal axis. A few days since, in reading over for the press a manuscript 

 which had been written last year, I observed I had drawn attention to 

 the circumstance that such relative equilibrium woidd be impossible if the 

 ellipsoid rotated round any other straight line. Almost immediately 

 afterwards I was accidentally glancing for the first time at the elaborate 

 treatise on Mechanics published in 1870 by Dr. TV. Schell, under the 

 title of " Theorie der Bewegung und der Kriifte," and I noticed an account 

 of Jacobi's theorem. Dr. Schell records that Jacobi was led to the dis- 

 covery of his theorem by reason of an erroneous statement, made by Pon- 

 tecoulant, that such a result was impossible; Jacobi undertook the 

 inquiry, as he said, by virtue of a "certain spirit of contradiction to 

 which he owed his most important discoveries " (see page 966 of Dr. 

 Schell's volume). It should be remarked, however, that as to the error, 

 Pontecoulant merely followed Lagrange. 



2. I was much surprised to find that on the same page Dr. Schell made 

 the following assertion : — " It has been lately shown by Dahlander that 

 the relative equilibrium of the rotating ellipsoid will subsist even when 

 the axis of rotation does not coincide with a principal axis of the ellip- 

 soid." A reference is supplied to a memoir by Dahlander in Poggendorffs 

 ' Annalen,' vol. cxxix. (1866) p. 443. Notwithstanding this combination 

 of authority the assertion is incorrect, as I shall now show. 



3. I assume that when a mass of fluid is rotating with uniform angular 

 velocity round a fixed axis, the problem of determining the pressure of 

 the fluid and the form of the free surface may be changed from a dyna- 

 mical form to a statical in the following manner : — Suppose the rotation 

 stopped, and supply at every point an acceleration at right angles to the 

 axis outwards from the axis, equal to ra> 2 , where to is the angular velocity, 

 and r is the distance of the point from the axis of rotation. This can be 

 easily demonstrated, and is, in fact, always taken for granted bv writers 

 on the subject. 



4. I will confine myself for simplicity to the case in which the assumed 

 axis of rotation passes through the centre of the ellipsoid ; but it will be 

 easily seen that the process is applicable when this condition is not ful- 

 filled. Suppose that an ellipsoid, of which the axes are 2a, 2b, 2c, rotates 

 about a diameter which makes with the principal axes angles whose direc- 

 tion cosines are I, m, n. Take for the equation to the ellipsoid 



