89] 
ON THE ATTRACTION OF ELLIPSOIDS (jACOBl’s METHOD). 
513 
And then writing 
X = aX,+ a'Y 1 + a."Z 
Y = /3X 1 + /3'Y 1 + /3"Z 1 , 
Z = r yX 1 + 7 / F 1 + 7 ^i> 
X= aX,+ a% + a"Z 1} 
r = $Xi+ &% + &%, 
Z = yXr + 7' Fi + 7 "Z 1 , 
if X, Y, Z are the cosines of the inclinations of a line PQ to the principal axes of 
the ellipsoids, X 1} Y u Z 1 will be the cosines of the inclinations of this line to the 
principal axes of the cone having P for its vertex, and circumscribed about the interior 
ellipsoid. In like manner, X, Y, Z being the cosines of the inclinations of a line PQ 
to the principal axes of the ellipsoids, X 1} Y u Z x will be the cosines of the inclinations 
of this line to the principal axes of the cone having P for its vertex and circumscribed 
about the exterior ellipsoid. Assuming that the points Q, Q are situated upon the 
exterior and interior ellipsoids respectively, suppose that X 1} Y u Z x and X ly Y } , Z^ are 
connected by the equivalent systems of equations, 
y -X /( Xf Yj 2 Z? 
X, = -u)^ + — + — r 
Yi = 
Z, = 
u — q 
u - q 
Y 1} 
U —) Z lt 
u—r 
X x = *J(u — u) 
Y 2 Z } 2 
+ + 
u — u u — q u — r) ’ 
Y - Y 
“V \u-q) *'■ 
then it will presently be shown that the points Q, Q are corresponding points, which 
will prove the geometrical theorem of Jacobi. Before proceeding further it will be 
convenient to notice the formulae 
a 2 b 2 c 3 _u — u 
f+ u g + u h + u af 
Xa Yb Zc u — u i Xjoq FA ^ Z x c Y \ 
f+u g + u h + u a 2 \u — u u — q u — r)' 
/ Xa Yb Zc y u — u! X 2 Y 2 ^ Z 2 \ 
\f +u g + u^ h + u) ^ a 2 \f + g + u h + u) 
= u-u i X 2 Jl_ + _Zl_\ = X 2 . 
a 2 \u — u u — q u — r) af ’ 
c. 
65