of homegeneous Ellipsoids. 
35 * 
position that one of the indeterminate angles is constant ; thus, 
if ip be constant, then dy = k cos. p cos. 4 . dp : and, because y 
must be constant when z varies, we must make 
dz == k" cos. c p sin. 4 • dp k" sin. p cos. 4 • dip 
o = k' cos. cp cos. 4 .dtp — k' sin. cp sin. 4 • dip, 
and, by exterminating dp, we get dz = k -~-^ . dip. Thus, by 
substitution, the formula (2) will become 
A — k' k”Jf sin. p cos. 4 .dp .dip . j — — ^7 J ; (3) 
and, 
A = | ( a^—k cos. p )*-{- ( b—k ' sin. p cos. 4 ) e + (c—k" sin. p sin. 4 ) e } i 
A'= | [a-\-k cos. p )'-f- (b—k' sin. p cos. 4 ) 2 + (c—k" sin. p sin. 4 )* } k 
the double fluent must be taken from p = 0, to p = -^ (n r de- 
noting half the pheriphery of the circle, whose radius is 1), 
and from 4 = 0, to 4 = S'ar. 
To obtain a further transformation of the last expression of 
A , we are now to determine the semi-axes of an ellipsoid, 
whose surface shall pass through the attracted point, and which 
shall have the same excentricities, and its principal sections 
in the same planes, as the given ellipsoid. Let li, h! , h" be the 
semi-axes required : then, because the attracted point is to be 
in the surface of the solid, 
a 1 , b z , c z 
T z "T“ IE T 1 : 
and, because the excentricities must be equal to those of the 
given ellipsoid, therefore h' x — /i* = k /2 — li = o', and h" x — h % 
= k" x — k* — e '* : hence 
b z 
T* + 
+ 
= 1; 
b z + e 2, 1 /j 1 -!- e' 5 
an equation which now contains only one unknown quantity,, 
Z z 2 
