1

–––––––

0.045505251

prism 411

17.357 and r(1,000,000,000) equals ≈

19.667. As 2.31 is approximately the natural

LOGA

-

RITHM

of 10 this suggested to Gauss that r(n) is behav-

ing as the natural logarithm function. He conjectured:

for large values of n, but was never able to prove this

claim.

One hundred years later J

ACQUES

H

ADAMARD

(1865–1963) and C

HARLES

-J

EAN DE LA

V

ALLÉE

-P

OUSSIN

(1866–1962) simultaneously, but independently, proved

Gauss’s conjecture using sophisticated techniques from

analytic

NUMBER THEORY

. The result is known today as

the prime number theorem. It tells us that the nth prime

number has value approximately n· ln(n).

principal See

INTEREST

.

principal axes The most general equation in three

variables of second degree is:

Ax2+ By2+ Cz2+ Dxy + Exz + Fyz

+ Gx + Hy + Iz + J= 0

with constants Athrough Fnot all zero. In three-dimen-

sional space, the graph of such an equation is a surface

called a quadric surface. Mathematicians have shown

that, by rotating and translating the coordinate axes, it

is possible to rewrite the equation with respect to a new

set of axes to simplify the form of the equation, reduc-

ing it to one of 13 different types. The new coordinate

axes are called the principal axes for the quadric.

Six of the possible forms of the equation lead to

nondegenerate quadrics:

1. Ellipsoid: + + = 1

2. Hyperboloid of one sheet: + – = 1

3. Hyperboloid of two sheets: + – = –1

4. Elliptic cone: + =

5. Elliptic paraboloid: + =

6. Hyperbolic paraboloid: – =

The remaining seven possible equations are degen-

erate quadrics:

1. Elliptic cylinder: + = 1

2. Hyperbolic cylinder: – = 1

3. Parabolic cylinder: =

4. Pair of planes: = or = 1

5. Single plane: = 0

6. Line: + = 0

7. Point: + + = 0

In two-dimensional space, with suitable rotation

and translation of the coordinate axes, any nondegen-

erate quadratic equation Ax2+ By2+ Cxy + D= 0 can

be rewritten as the equation of either an ellipse, a

hyperbola, or a parabola, or, in the degenerate cases, a

pair of lines, a single line, or a point. The axes in the

new coordinate system are again called principal axes.

See also

CONIC SECTIONS

.

prism Any

POLYHEDRON

with two faces (the bases)

that are congruent polygons lying in parallel planes and

such that the remaining faces (the lateral faces) are par-

allelograms is called a prism. Specifically, a prism is a

CYLINDER

with a polygonal base.

The lines joining the corresponding vertices of the

base polygons of a prism are called lateral edges. If the

lateral edges of a prism meet its bases at right angles,

then the prism is called a right prism. A prism that is

not right is called oblique.

Prisms are named according to their bases. A trian-

gular prism has two triangular bases (and three lateral

faces); a quadrangular prism has two quadrilateral

bases and four lateral faces. A

CUBE

is an example of a

right quadrangular prism.

z2

–

c2

y2

–

b2

x2

–

a2

y2

–

b2

x2

–

a2

x2

–

a2

x2

–

a2

y2

–

b2

x2

–

a2

y

–

b

x2

–

a2

y2

–

b2

x2

–

a2

y2

–

b2

x2

–

a2

z

–

c

y2

–

b2

x2

–

a2

z

–

c

y2

–

b2

x2

–

a2

z2

–

c2

y2

–

b2

x2

–

a2

z2

–

c2

y2

–

b2

x2

–

a2

z2

–

c2

y2

–

b2

x2

–

a2

z2

–

c2

y2

–

b2

x2

–

a2