[604 
604] 
ELLIPSOIDAL SHELL ON AN EXTERIOR POINT. 
309 
13. The expression for shows that R is the radius vector, cosine-inclinations 
a, ¡3, 7, in an ellipsoid semi-axes F, G, H, which may be regarded as having its centre 
at Q; viz. this is the “ auxiliary ellipsoid.” And this being so, we have 
p" _ n 
r"~ R' 
P_ 
r 
It appears from these equations that, drawing from Q parallel to PR" a line 
QM, = 12, and from its extremity M parallel to PQ a line to meet QR' in T, the 
locus of T is the auxiliary ellipsoid. 
14. By what precedes, the angles RPQ, R"PQ are equal to each other, say each 
is = </>; the triangle R'PR" gives 
that is, 
viz. this is 
= R*(B»-AC)] 
or say 
cos <£ = R VB 2 — AG; 
a remarkable equation which may also be written 
cos 0 = § ■ £ O' + r"), 
if, as before, v is the semi-diameter parallel to R'R". 
In virtue of the equation A = 
In virtue of the equation A = - which 
1 tJlP-AC 
, mR 
cos <f> — j 
which defines A, the equation becomes 
and we thus complete the demonstration of the several geometrical theorems upon 
which the investigation was founded.