120 On an Erroneous Extension of Jacobi's Theorem. [Jan. 16, 



Let O denote the centre of the ellipsoid, M any point of the mass whose 

 coordinates are x, y, z, and N the point where the perpendicular from 

 M on the axis of rotation meets that axis. Let £, r/, '( be the coordinates 

 of N. 



Then the acceleration w 2 MN, when resolved parallel to the axes, gives 

 rise to the components 



«\x-i\ uXy-n), u>\z-0 

 respectively. We must now determine £, 77, £. 



"We know that 



cosMON=^+^±^ ; 



OM 



hence OMcos MON, that is 01$=lx+my-\-nz. Denote this by v ; then 

 we have 



%=lv, rj=mv, l — nv. 

 The attractions of the elhpsoid at (x, y, z) parallel to the axes will be re- 

 spectively 



Vx, Qy, Bz ; 



where P, Q, B are certain constants, in the form of definite integrals, 

 which depend on the ratios of the axes of the elhpsoid. 



Hence if p denote the pressure, and p the density of the fluid, we have 



^.=p{ ( o\x-lv)-Vx}, 



(XX 



y- = / J { w 2 (z — nv) — Jlz } . 



Therefore the form of the free surface will be determined by the equation 

 w 2 (x 2 + if + z'*)-ioXlx + my + nzy-Vx*-Qy*-~Rz*= constant. . (2) 

 By supposition, (1) and (2) must represent the same surface. But this 

 is obviously impossible, unless two out of the three I, m, n vanish. Thus 

 the axis of rotation must coincide with one of the principal axes ; and 

 then it follows in the known way that this must be the least principal axis. 



5. Now let us turn to the memoir by Dahlander in Poggendorff's 

 1 Annalen.' The process is this. Dahlander supposes that there are 

 three simultaneous angular velocities, w, w', w", round the axes of x, y, z 

 respectively ; and then he assumes the equation 



^=_(P_ w ' 2 _ w ''>YZ ( r-(Q- W a -a;'%^-(B- W ' 2 - W 2 )^fo. 



P 



This equation, however, is unjustifiable. Dahlander does not say how he 

 obtained it, so that it is impossible to point out exactly where his error lies. 

 Perhaps the equation was supposed to hold in virtue of some unwarranted 

 extension of the principle in article 3. To show that the equation is 



wrong, it is sufficient to observe that it makes ^ involve only the 



