228 The Sphere and least Circumscribing Cone. 
Putting the first = 0, and reducing, we find x = 3r; therefore 
the height of the cone is 4r, or twice the height of the sphere. 
akg : ar?(e+r)? 
By substituting 3r for z in the expression for the cone, 3(a2 =r)’ 
it becomes, when reduced, 3° (2r)*. But the sphere is 5° (2r)*. 
Therefore the volume of the cone is twice that of the sphere. 
oe? : ; pie 
The same substitution being made in the value of dpe’ it be- 
1 ae 
comes +~> indicating that the cone is a minimum, which is in- 
deed sufficiently evident from the nature of the question. 
Another proportion on the triangles ABC and ADE, gives the 
; UAe+r P 
slant height = ——=———- And since the radius of the base is 
Vv 7? —fr? 
r(r+r ' A + 
~ the circumference is psa uell ; and therefore the con- 
Vg? —yi Ve? —p? 
ara(@+r)? zra(¢+r) 
PoaanSertt. aes ; 
ar? (e+r 
£ 
ay )) gives for the entire surface of the cone, 
vex surface = This added to the base, 
(which equals 
xr al 2 r 
pin * The first and second differentials, after dividing by 
ds 2 Ay) 2 
the constant <r, are, a= alee (+r) Apes dedi 
a? 
2 
“$s 
2 
Sr? 
(r-r) As these are precisely like the former, they lead to the 
same conclusion, that 7 equals 3r, and corresponds to a minimum 
value of the function. Therefore, the circumscribing cone of 
least surface also, has twice the altitude of the sphere, and is 
identical with the cone of least volume. T'o obtain the amount 
end oe 
t-r 
of surface, replace x by 3r in the expression 
comes 27(2r)?. Now the surface of the sphere is (2r)?. 
Wherefore the cone of least surface about a sphere, has twice 
the surface of the sphere. 
hese results brought together are as follows: 
7 phere, Min. Cir. Cone. 
Altitude, - “ Qp m ~ dr 
Surface, 8 4 en(Qpys ape 
Bs Velunte t +  cuggRe prin © 2 se g(r)" 
eed Meet eC aeett 
