Download - Mathematical Description
![Page 1: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/1.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 1/109
Mathematical description[edit] quation[edit]
tenaries for different values of a
ree different catenaries through the same two points, depending horizontal force being and λ mass per unit length.
e equation of a catenary in Cartesian coordinates has the form[27]
where cosh is the hyperbolic cosine function. All catenary curves are similar to each other, having eccentricity = √2.
Changing the parameter a is equivalent to a uniform scaling of the curve.[30]
The Whewell equation for the catenary is[27]
Differentiating gives
and eliminating gives the Cesàro equation[31]
The radius of curvature is then
which is the length of the line normal to the curve between it and the x -axis.[32]
Relation to other curves[edit]
![Page 2: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/2.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 2/109
When a parabola is rolled along a straight line, the roulette curve traced by its focus is a
catenary.[33]
The envelope of the directrix of the parabola is also a catenary.[34]
The involute from the
vertex, that is the roulette formed traced by a point starting at the vertex when a line is rolled on a
catenary, is the tractrix.[33]
Another roulette, formed by rolling a line on a catenary, is another line. This implies that square
wheels can roll perfectly smoothly if the road has evenly spaced bumps in the shape of a series of
inverted catenary curves. The wheels can be any regular polygon except a triangle, but the catenary
must have parameters corresponding to the shape and dimensions of the wheels.[35]
Geometrical properties[edit]
Over any horizontal interval, the ratio of the area under the catenary to its length equals a, independent
of the interval selected. The catenary is the only plane curve other than a horizontal line with this
property. Also, the geometric centroid of the area under a stretch of catenary is the midpoint of the
perpendicular segment connecting the centroid of the curve itself and the x-axis.[36]
Science[edit]
A charge in a uniform electric field moves along a catenary (which tends to a parabola if the charge
velocity is much less than the speed of light c ).[37]
The surface of revolution with fixed radii at either end that has minimum surface area is a catenary
revolved about the x-axis.[33]
Analysis[edit] Model of chains and arches[edit]
In the mathematical model the chain (or cord, cable, rope, string, etc.) is idealized by assuming that it is
so thin that it can be regarded as a curve and that it is so flexible any force of tension exerted by the
chain is parallel to the chain.[38] The analysis of the curve for an optimal arch is similar except that the
forces of tension become forces of compressionand everything is inverted.[39]
An underlying principle is
that the chain may be considered a rigid body once it has attained equilibrium.[40]
Equations which define
the shape of the curve and the tension of the chain at each point may be derived by a careful inspection
of the various forces acting on a segment using the fact that these forces must be in balance if the chain
is in static equilibrium.
Let the path followed by the chain be given parametrically by r = ( x , y ) = ( x (s), y (s))
where s represents arc length and r is the position vector . This is the natural parameterizationand has
the property that
where u is a unit tangent vector .
![Page 3: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/3.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 3/109
Diagram of forces acting on a segment of a catenary from c to r . The forces are the tension T0 at c, the
tension T at r , and the weight of the chain (0, −λgs). Since the chain is at rest the sum of these forces must be
zero.
A differential equation for the curve may be derived as follows.[41]
Let c be the lowest point on the
chain, called the vertex of the catenary,[42]
and measure the parameter s from c. Assume r is to the
right of c since the other case is implied by symmetry. The forces acting on the section of the chain
from c to r are the tension of the chain at c, the tension of the chain at r , and the weight of the chain.
The tension at c is tangent to the curve at c and is therefore horizontal, and it pulls the section to theleft so it may be written (−T 0, 0) where T 0 is the magnitude of the force. The tension at r is parallel to
the curve at r and pulls the section to the right, so it may be written T u=(T cos φ, T sin φ), where T is
the magnitude of the force and φ is the angle between the curve at r and the x -axis (see tangential
angle). Finally, the weight of the chain is represented by (0, −λgs) where λ is the mass per unit
length, g is the acceleration of gravity and s is the length of chain between c and r .
The chain is in equilibrium so the sum of three forces is 0, therefore
and
and dividing these gives
It is convenient to write
which is the length of chain whose weight is equal in magnitude to the tension
at c.[43]
Then
is an equation defining the curve.
The horizontal component of the tension, T cos φ = T 0 is constant and the
vertical component of the tension, T sin φ = λgs is proportional to the length of
chain between the r and the vertex.[44]
Derivation of equations for the curve[edit]
The differential equation given above can be solved to produce equations for
the curve.[45]
From
the formula for arc length gives
Then
![Page 4: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/4.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 4/109
and
The second of these equations can be integrated to give
and by shifting the position of the x -axis, β can be taken to
be 0. Then
The x -axis thus chosen is called the directrix of the
catenary.
It follows that the magnitude of the tension at a
point T = λgy which is proportional to the distancebetween the point and the directrix.
[44]
The integral of expression for dx /ds can be found
using standard techniques giving[46]
and, again, by shifting the position of the y -axis, α
can be taken to be 0. Then
The y -axis thus chosen passes though the
vertex and is called the axis of the catenary.
These results can be used to eliminate s giving
Alternative derivation[edit]
The differential equation can be solved
using a different approach.[47]
From
it follows that
and
Integrating gives,
![Page 5: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/5.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 5/109
and
As before,
the x and y -axes can
be shifted so α and β
can be taken to be 0.
Then
and taking the
reciprocal of both
sides
Adding and
subtracting
the last two
equations
then gives the
solution
and
Determiningpar ameter s[edit]
In
gener al the
para
meter
a an
d the
positi
on of
the
axisand
direct
rix
are
not
![Page 6: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/6.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 6/109
given
but
must
be
deter
mine
d
from
other
infor
matio
n.
Typic
ally,
the
infor
matio
n
given
is
that
the
caten
ary is
susp
ende
d at
given
point
s P 1
and
P 2 an
d
with
given
lengt
h s.
The
equat
ion
can
be
deter
mined in
this
case
as
follow
![Page 7: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/7.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 7/109
s:[48]
Rela
bel if
nece
ssary
so
that
P 1 is
to the
left
of P 2
and
let h
be
the
horiz
ontal
and v
be
the
vertic
al
dista
nce
from
P 1 to
P 2. Tr
ansla
te the
axes
so
that
the
verte
x of
the
caten
ary
lies
on
the y-
axis
and
itsheigh
t a is
adjus
ted
so
![Page 8: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/8.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 8/109
the
caten
ary
satisfi
es
the
stand
ard
equat
ion of
the
curve
a
n
d
le
t
t
h
e
c
o
o
r
d
i
n
a
t
e
s
o
f
P
1
a
n
d
P
2
b
e
(
x
1,
![Page 9: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/9.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 9/109
y
1)
a
n
d
(
x
2,
y
2)
r
e
s
p
e
c
ti
v
e
ly
.
T
h
e
c
u
r
v
e
p
a
s
s
e
s
t
h
r
o
u
g
ht
h
e
s
e
![Page 10: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/10.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 10/109
p
o
i
n
t
s
,
s
o
t
h
e
d
if
f
e
r
e
n
c
e
o
f
h
e
i
g
h
t
is
![Page 11: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/11.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 11/109
![Page 12: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/12.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 12/109
![Page 13: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/13.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 13/109
![Page 14: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/14.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 14/109
![Page 15: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/15.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 15/109
![Page 16: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/16.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 16/109
![Page 17: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/17.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 17/109
![Page 18: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/18.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 18/109
![Page 19: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/19.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 19/109
![Page 20: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/20.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 20/109
![Page 21: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/21.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 21/109
![Page 22: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/22.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 22/109
![Page 23: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/23.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 23/109
![Page 24: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/24.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 24/109
![Page 25: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/25.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 25/109
![Page 26: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/26.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 26/109
![Page 27: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/27.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 27/109
![Page 28: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/28.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 28/109
![Page 29: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/29.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 29/109
![Page 30: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/30.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 30/109
![Page 31: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/31.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 31/109
![Page 32: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/32.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 32/109
![Page 33: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/33.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 33/109
![Page 34: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/34.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 34/109
![Page 35: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/35.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 35/109
![Page 36: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/36.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 36/109
![Page 37: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/37.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 37/109
![Page 38: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/38.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 38/109
![Page 39: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/39.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 39/109
![Page 40: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/40.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 40/109
![Page 41: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/41.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 41/109
![Page 42: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/42.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 42/109
![Page 43: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/43.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 43/109
![Page 44: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/44.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 44/109
![Page 45: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/45.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 45/109
![Page 46: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/46.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 46/109
![Page 47: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/47.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 47/109
![Page 48: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/48.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 48/109
![Page 49: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/49.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 49/109
![Page 50: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/50.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 50/109
![Page 51: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/51.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 51/109
![Page 52: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/52.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 52/109
![Page 53: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/53.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 53/109
![Page 54: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/54.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 54/109
![Page 55: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/55.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 55/109
![Page 56: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/56.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 56/109
![Page 57: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/57.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 57/109
![Page 58: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/58.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 58/109
![Page 59: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/59.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 59/109
![Page 60: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/60.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 60/109
![Page 61: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/61.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 61/109
![Page 62: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/62.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 62/109
![Page 63: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/63.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 63/109
![Page 64: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/64.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 64/109
![Page 65: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/65.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 65/109
![Page 66: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/66.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 66/109
![Page 67: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/67.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 67/109
![Page 68: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/68.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 68/109
![Page 69: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/69.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 69/109
![Page 70: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/70.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 70/109
![Page 71: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/71.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 71/109
![Page 72: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/72.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 72/109
![Page 73: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/73.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 73/109
![Page 74: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/74.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 74/109
![Page 75: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/75.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 75/109
![Page 76: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/76.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 76/109
![Page 77: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/77.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 77/109
![Page 78: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/78.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 78/109
![Page 79: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/79.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 79/109
![Page 80: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/80.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 80/109
![Page 81: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/81.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 81/109
![Page 82: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/82.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 82/109
![Page 83: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/83.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 83/109
![Page 84: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/84.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 84/109
![Page 85: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/85.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 85/109
![Page 86: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/86.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 86/109
![Page 87: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/87.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 87/109
![Page 88: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/88.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 88/109
![Page 89: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/89.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 89/109
![Page 90: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/90.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 90/109
![Page 91: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/91.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 91/109
![Page 92: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/92.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 92/109
![Page 93: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/93.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 93/109
![Page 94: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/94.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 94/109
![Page 95: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/95.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 95/109
![Page 96: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/96.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 96/109
![Page 97: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/97.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 97/109
![Page 98: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/98.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 98/109
![Page 99: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/99.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 99/109
![Page 100: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/100.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 100/109
![Page 101: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/101.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 101/109
![Page 102: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/102.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 102/109
![Page 103: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/103.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 103/109
![Page 104: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/104.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 104/109
![Page 105: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/105.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 105/109
![Page 106: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/106.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 106/109
![Page 107: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/107.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 107/109
![Page 108: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/108.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 108/109
![Page 109: Mathematical Description](https://reader031.vdocuments.us/reader031/viewer/2022021323/577cc6b31a28aba7119ef2ee/html5/thumbnails/109.jpg)
8/12/2019 Mathematical Description
http://slidepdf.com/reader/full/mathematical-description 109/109