122 
Mr. Whewell 07 i calculating 
in what order the faces occur, and which are adjacent. This 
may be done in the following manner : 
Let A I fig. 21, be any parallelepiped of which the edges 
Ax, Ay, Az are a, b, c. Let an ellipsoid be described, of 
which the center is I, touching three planes of this parallele- 
piped in D, E, F. If we suppose any secondary plane, de- 
duced from this parallelepiped, to be drawn so as to touch 
the ellipsoid in P, the situation of the points P will determine 
the position of the planes. Let A x + B;' + Cz = mhe the 
equation to the plane. The equation to the ellipsoid will be 
, {b—yT I jc—z)^ 
"T T c* ^ ■ 
And that the plane may touch the ellipsoid, the difi'erential 
co-efficients and |^J must be the same in both. Hence 
/dy't A 6* ^ («— ■^') . 1^^] ^ (ff— .y) 
B {b — y) ’ \dx) C (c — z) ‘ 
Therefore 
a — X b — y 
~A~^ TF 
a — X c — z 
A a* Cc^ ■ 
And substituting in the equation to the ellipsoid we have 
{a x) (a xj -|- {a 
A a 
—x) 
a 
X 
V (A^ c‘) 

y — V (A^ c^) 
and c — z=^ 
Cc 
V (A» d^ + B^ 6^ + d) 
Knowing the position of the points P for all the planes, we 
have the polyhedron, on the supposition that it is made such 
that the ellipsoid can be inscribed in it ; which is always pos- 
sible by supposing the planes to move parallel to themselves 
till they touch it. 
We shall see more clearly the position of the points P if 
