512 
ON THE ATTRACTION OF ELLIPSOIDS (jACOBl’s METHOD). 
[89 
pQ forme avec deux des axes du second cône ; ces deux axes sont les tangents situés 
respectivement dans les sections de plus grande et de moindre courbure de chaque 
ellipsoïde, le troisième axe étant la normale à l’ellipsoïde.’ Tout cela semble difficile 
à établir par la synthèse.” 
The object of this paper is to develope the above method of finding the attraction 
of an ellipsoid. 
Consider an exterior ellipsoid, the squared semiaxes of which are f+ u, g + u, h + u ; 
and an interior ellipsoid, the squared semiaxes of which are f+u, g + u, h + u. Let 
a, p, q be the elliptic coordinates of a point P on the exterior ellipsoid, the elliptic 
coordinates of the corresponding point P on the interior ellipsoid will be u, p, q, and 
if a, h, c and a, b, c represent the ordinary coordinates of these points (the principal 
axes being the axes of coordinates), we have 
a 2 = 
6 2 = 
c- = 
(f+ u ) (f+q) (f+r) 
if-9) (f~ h ) 
(.9 + u ) (9 + g) (9 + r) 
(9 - h ) (9 ~f) 
(h + u) (h + q) (h + r) 
(h-g)(h-f) 
= (f+u) (f+q) (f+r) 
(f~9)(f~h) 
№-(9 + û) (g + q){g + r ) 
(9 - h ) (9 ~f) 
—2 _ (■h + u) (h + q)(h + r) 
(h~f) (h-g) 
I form the systems of equations 
2 _ O+/) (u + g)(u + h) 
(u — q)(u — r) 
h 2 = (g +/) (9 + 9) (g + Æ ) 
(q -r){q- u) 
r 2^( r +f) ( r + 9) (r+h) 
(r — u)(r — q) 
d 2 = (T+/K u +9)Jp+ h) 
(ü — q) (ü — r) 
12 = (g +/) (g + 9) (9 + h) 
(g ~ r )(q- ü) 
?i a = ( r +/) 0 + 9) (r + h) 
(r — u) (v — q) 
a x a 
'f+u’ 
/3 
af 
9 + u> 
<s> 
1 + 
-si 
II 
b x a 
f+ g ’ 
/3' 
bj> 
~ g + q’ 
/ ÔjC 
7 + 
c x a 
'f+r' 
/3" 
cj) 
~ g + r’ 
7 A + r’ 
a+i 
f+u’ 
35 
af> 
~ 9 + u’ 
oqc 
7 A + à ’ 
'f+q* 
bf> 
~ g + q’ 
“/ \c 
7 = A + ?’ 
C+JL 
f+r ’ 
r 
cf> 
~ g + r’ 
~n C-fi 
7 h + r‘