( 575 ) 
and the right line cut by it 
(a2 =e a= ala, tit (bi Seb ij  « . (10) 
lying in the plane (8). 
If in (7) we substitute 29, ue, re for w,, y,, 2, and reduce it to 
the form 
a? GC — i) (vy — uz) +b? (2 — «) (Az — va) + ¢? 5 — v ius —Ay)=0, 
Ox 9 9 
then @ = furnishes the equation 
(a? — ?) Auz + (b? — c?) uve + (ce? —a')vdy=0, . . (LI) 
which represents the plane containing the normals with the direction 
(4, u,v). The footpoints of these normals lie evidently on the right line 
DELEN SUP = SOT 6 oe at a eee (UI) 
6. We determine with respect to the ellipsoid (4,) the polar line 
of the normal n, having P, as footpoint. 
For an arbitrary point P' of that normal we find the polar plane 
at a 
4 
=> 
3 a 
7 zj 
20 — es 
For all values of w' this plane passes through the line of inter- 
section of the planes 
Er SE = Boia ate! gatas) 
a b c 
et be EL) 
a b c 
This line of intersection is the required polar line. When /, changes 
it displaces itself evidently parallel to itself. 
Out of (13) and (14) we obtain the equation 
(aise) nd (OS CUBE Fe 
bt c' a}. 
which becomes identical with the equation 
rg — de lie 
Ys 23 
of a projecting plane of the normal n, with footpoint P,, if the 
conditions are satisfied 
(a*—b?*) y, 6? (a?—c’) z, ANP Ce 
Eo (Gacy, | be 1 
From this we can deduce that the polar line of the normal n, 
with respect to the surface (/,) is again a normal n,; the footpoints 
vi 
, and P, are connected with each other by the involutory relations 
