120 M. G. R. Dahlander on a New Species of Figures 



system, whose origin is in the centre of the ellipsoid. It has 

 now to be shown that an ellipsoid of rotation can form the limit- 

 ing surface for the cavity within the fluid, supposing the pres- 

 sure is not directed against the cavity ; or if such is the case, 

 that it is counterbalanced by the pressure of a body of gas 

 enclosed in the cavity, whose density is, however, so small that 

 the attraction exerted by the gas on the fluid may be neglected. 

 Let X, Y, Z denote the three components parallel with the co- 

 ordinate axes of the attraction which the ellipsoid and the fluid 

 exercise on a particle of the fluid having the coordinates x, y, z. 

 The attraction of the mass of the hollow ellipsoid on the point 

 x, y t z is equal to that of a solid ellipsoid, diminished by that 

 which another solid ellipsoid equal to the hollow part would 

 exert on the point. Let the components of attraction of the 

 former ellipsoid be 



— Mx, —My, — Ns, 



and those of the latter 



-Mx, -Wy, -Wz. 



The attraction of the fluid, which is supposed to be bounded by 

 the two spheroids, is equal to that which an ellipsoid of massive 

 fluid would exercise, diminished by that of an ellipsoid with the 

 same density corresponding to the inner cavity. Let the com- 

 ponents of the former attraction be 



-M'fc, -M"y, -N"*, 

 and those of the latter 



-W'x, -M"fy, -N'"*. 

 Then we get 



X=(-M+M'~M'' + M'>n 



Y=(-M + M'-M'' + M''')y, I . . (1) 



Z=r(-N + N'-N" + N'yJ 



The differential equation of the surfaces de niveau will there- 

 fore be 



(-M + M-M" + M'"){xdx + ydy) + (-N + W-W + W")zdz 



+ w\xdx + ydy)=0 } (2) 



where w is the constant angular velocity. 



M, M', M", N, N', N" are independent of x } y, z; but M" ; 

 and N" ; are generally dependent upon them. When the in- 

 nermost bounding surface is alone considered, even M'" and W" 

 become independent of x, y, z. If this surface be a spheroid, 

 it will be possible to express its equation in the form 



t±t + z l -1 ... (3) 



a 2 + a*(l+\" 2 ) -1 K) 



