THE 



LONDON AND EDINBURGH 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



T 



[THIRD SERIES.] 



AUGUST 1838. 



XII. On the Co7iditions of Equilibrium of a Homogeneous 

 Planet in a Fluid State. By James Ivory, X'.fl'., F.R.S., 



HIS problem is treated in the Traite de Mecanique of 

 Poisson, a work which is in everybody's hands ; and, in 

 order to abridge, I shall take the differential equation of a 

 level surface from that work (Equat. (b), p. 536, vol. ii. edit. 

 2nd,) viz. 



X.dx-\-Ydy-\-Zdz-\-a^{xdx + ydy) = 0: 



here ^, 3/, z are the rectangular coordinates of a point in the 

 level surface, z being parallel to the axis of rotation : X, Y, 

 Z, are the attractions of the whole mass upon the particle, in 

 the respective directions of cc,y,z'. and a is the angular velo- 

 city of rotation about the axis. By integrating the foregoing 

 equation we obtain, 



Const. = r^Kdx + Ydy -^Zdz)+ ^ G^'+/) ... (1.) 



The whole mass being divided into two parts by ^he level, 

 surface, I shall put P, Q, R, for the attractive forces parallel 

 to ^, y, z, which the matter within the level surface exerts 

 upon the particle in that surface ; and P', Q', R' for the like 

 attractions of the matter without the level surface : so that we 

 shall have 



X = P + P', Y = Q + Q', Z = R + R'; 



and the equation of the level surface will be 



* Communicated by the Author. 

 Phil. Mag, S. 3. Vol. 13. No. 80. Aug. 1838, G 



