AND THE FIGURE OF A HOMOGENEOUS PLANET. 
139 
d.(V (r) - V'(r)) Ox d.(V(r) - V») dy d . ( V (r) - V' (r)) ^ 
dtx G?S "l - Jz/ ' dz rfs 
and this resultant must be equal to zero in whatever direction d s is drawn, if 
the attraction of the stratum upon a m be perpendicular to the surface ab c. 
Wherefore we have, 
d. (y (r) — V'(r) > ) d . (y (r) — y (r)} d . ( V (r) — V (r)') 
+ Z d2/+ _^_ l d%=0i ( 5) 
and, by integrating, 
V (r) — V' (r) = Constant, (6) 
which equation must be true at every point in the surface a b c. 
Again, the attractive forces of the interior mass urging the molecule am 
inwards in the direction of x, y, z, are respectively equal to 
d . y' (r) _ d.y' ( r ) d . V ( r ) 
dx ’ dy ’ dz 
Let f denote the centrifugal force at the distance unit from the axis of rota- 
tion ; and, the distance of am from the same axis being y 2 + z 2 , the centri- 
fugal force of the particles of am will be f^Jy 1 + z 2 ; and the resolved parts 
of this force acting in the prolongations of y and z, will be fy and f z. Where- 
fore the total accelerating forces urging am in the directions of x, y, z, and 
tending to shorten these lines, are respectively, 
W -{*$*+/>)■ (*£*+/»)■ 
and, the condition that the resultant of these forces is perpendicular to the sur- 
face ab c, is expressed by this differential equation. 
d . y' (r) 
dx 
dx- ( ^dnr + fy) d y - C*'L (f) +/*) dz = o 
(7) 
In the equations (5) and (7) the forces expressed by the co-efficients of the 
differentials, act on the same particles and have opposite directions in the 
same lines ; wherefore by subtracting the former from the latter, we have, 
t 2 
