The times. [volume] (Richmond, Va.) 1890-1903, 1897-05-09

be fastened ct the point c, and the horse
tied at the end T. keeping the rope tehi'
the horse carrying the end I around 10 n
until the rope is tangent to the tank
at c will describe the quarter OF n cir,
conference ia. Moving on. the rope VIII
lap around the tank and the horse IN ms
travel from a to NJ Will describe an m
volute to the arc en-. AS the horse moves
from A TO NJ the rope becomes tangent TO
every point in the arc en-. and the length
OF the tangent always equals the length
Of the arc from point of tangency 10
point NJ. Thus tangent AC equals arc
ON When the horse is at point the
rope will be tangent TO tank at point r.,
and the tangent FL will equal arc FM.
The half of the area over which the
horse can Craze will be the quarter OF
circle eta and the space included De-
tween the involute AM, tangent AC and
arc CM.
Suppose instead OF tho rope be-
ing one hundred feet In length
that II equals the circumference OF
the tank. and that the radius of
the tank equals /. Let one end of the
rope be fastened at NJ ana tne morse Or Or
fastened TO the free end and start from NJ
and unwind always S@eping the rope
tight. The horse will pass the points H.
49, i2. ets.until he arrives at a, descriD-
Ing exactly the same curve as when
passing from 10 NJ with a rope 100 feet
long The horse will finally arrive at the
point Y. having described a polar curve.
which is one spire of spiral The tan
gent MY will equal the circumference
OF the tank.
II is demonstrated in calculus that the
dfferential area of polar curve equals
the radius tVector squared. multiplied
the differential OF the angle), divided by
Draw the line 40. which will he the ra-
dius vector at point A. also draw 70.
which will be the radius vector at the
point Y. We now have two right angled
triangles. AGO and MYO.
Let the hypothenuse 40 equal and Of
equal b. Since OC and ON equal /, we
have the proportion squared is TO q
plus tangent &C squared, as b squared is
10 plus tangent MY squared. Tangent
AC equals arc CM. and arc CM measures
angle moo; therefore arc CM equals an
gle moo.
Tangent MY equals circumference CMO
equals pr multiplied by r. Since equals
] tangent MY equals pi and tangent MY
squared equals pi.
r squared is to plus angle squared az
b squared is to plus pi. or squared
times plus pi square equals b squared
times ] plus angle squared.
squared times plus pi squared divld-
ed by b squared equals plus angle squar-
ed
Angle squared equals squared times l
plus pi squared minus b squared, Ovid
Angle equals l divided by b times the
square root OF square times ] plus 4pi.
minus b square. differential Of the angle
equals l divided by b times rdr times
plus 4pi squared. divided by the square
root OF square times ! plus pi square
minus b squared.
Substituting in equation for differential
OF the arc OF a polar curve the value of
the differential OF the angle we have diff
ferential area equals divided by % times
cube dr. times plus 4pi squared. Ovid
ed BY the square root OF square times
plus 4pi squared. minus % squared.
A times the integral of divided by 2b
times cube DIR times ! plus 4pi squared.
divided by the square root OF squared
times plus 4pi square minus b equals the
integral OF square divided by 20 times
rdr times plus 4pi squared divided by
the square root OF square times 1 plus
4pi square minus b.
square divided by 2b times the square
root OF r square times l plus app square
minus b square. minus ] divided by
times the integral rdr times the square
root OF square times plus 4pi square.
minus b equals 4 square divided by n
times the square root OF square times
plus 4p squared minus b square minus I
square times ] plus 4pi square. minus D
square. 10 the three half power, divided
by 90 times l plus 4pi square.
Since the hypothenuse OF right angied
triangle equals the sum of the squares or
the two other sides we have In the TRAN
gle MOY squared equals | plus 4pi
squared and In triangle AGO squared
equals &C squared pluc co squared.
Since tangent &C equals and radius
co EQUALS ] We have square e0uals
squared plus l squared equals s. Suosu-
tuting I'm equation the value OF b square
and square We have square minus
square minus l. 20 the three half power
divided by s. Since square equals ,.
solving equation the area A equals y l-5
equals area OF involute to the arc Of cir
Gle at radius co or ! and tangent AC or
For radius 20 and Tangent 100 we have
squared is TO 2l-s as 50 squared is TO
area AMOC.
Since square equals We have AMOC
equals 80 squared times IL equals 900
times 2 is equals 5S33 10.
The area OF q rector OF a circle equals
length OF arc multiplied by half OF ra-
dius. therefore since arc CM equals 100
and half OF radius equals 2 We have area
OF sector COM EQUALS 900
5Sss Is minus $500 equals S.9Ss 10 SO. n.
EQUALS area included between tangent AC.
arc ON and involute AM.
Area OF quarter OF circle with diameter
CT zoo IT.. equals 1884
1854 plus ?. 2ss 1-3 equals ~ ISL Ia 50. feet.
EQUALS half OF area over which horse can
Craze. ll.lN Is times equals 292st 20
ga. feet. total area over which horse
can Craze.
Which Mas fheIargrs'?
In what navy Is the largest war YES
Fel? III. A.
In the italian nav ~ present. The
largest vessels are the Giovanni Bausan.
the Lepanto. Italia and Duilio. OF about
ts,0co tons each. But the British are
building larger vessels. and some five
IN number OF the ''Illustrious" class will
have nearly 1900 tons displacement each.
Exit Pleas
How may destroy feas and keep them
away? A. s.
oil OF pennyroysl u recommended az
4 hsaishsx Of 6sssl shaneuilo Sewers