of homogeneous Ellipsoids. 
37* 
ellipsoid rises to the third degree, is, in this case, depressed 
to a quadratic. In effect, the equation for h* when t 2 = e'*, 
becomes 
4 - 
whence 
and so 
b* + c* 
h* + e 1 
= 1, 
h" — (a 2 + b 2 -j- c 2 — e 2 ) h 2 = a 2 e% 
2 li 2 = a* + b* + c 2 _ e 2 4. vV 4- 6 2 + c 2 - e 2 ) 2 + 4a* e*. 
In the oblong spheroid, one of the semi-axes k' and k" must 
be made equal to the least semi-axis k, which corresponds to 
the suppposition of e n = o. In this case, the formulas (7) will 
become 
. p _dk 
A — - 3^M .J k3 ^ + 
B = SbM 
C = $cM .J p (4 . + 
In these expressions k is the radius of the equatorial circle of 
the spheroid, and not the semi-axis of revolution, which is = 
v/F+ e L \ and if we change k to denote the semi- axis of re- 
volution, which requires that V k 2 — e 2 be substituted for k ; 
and, for the sake of uniformity with the formulas for the ob- 
late spheroid, likewise interchange a and b } and A and B, in 
order that a may denote the ordinate parallel to the axis of 
revolution, and that A may express the attractive force in the 
same direction ; then, the last expressions will become 
A = 33 M .J 
B = S bM 
C = 3«M 
* Page 351. 
