The Sphere and least Circumscribing Cone. 227 
others; the energetic and independent public man, untiring in 
his energy, and sacrificing his private means in the public cause; - 
or the universal instructor of youth and age, the source of as fre- 
quent and general reference as the pages of a cyclopedia. 
The soldier dies honored who falls in battle. He too, perished 
“on the field of his fame”—a field whose victories are bloodless, 
and in whose fruits, untainted by misery and crime, the whole 
human family may rejoice. 
Art. XXV.— The Sphere and least Circumscribing Cone ; by 
Prof. E. 8. Sneux, of Amherst College. 
In searching for practical examples of maxima and minima, for 
my pupils in the Differential Calculus, I have fallen upon the fol- 
lowing relations of the sphere to its circumscribing cone, which 
are very singular, and to me entirely new. : 
. e cone of least surface, which cireumscribes a given 
sphere, is also the cone of least vo/wme, circumscribing the same 
her 
e. 
2. This minimum cone has twice the height of the sphere, 
twice its surface, and twice its volume. ; 
3. The base of the same cone equals the surface of the hemi- 
sphere ; and the convex surface of the cone, three times the sur- 
e of the hemisphere. ' 
Demonstration. 
In the accompanying sectional figure, 
which it is unnecessary to describe, let AC 
= z,and BC or CD =r; then the height 
of the cone is z+r. The similar triangles, 
3 r(x+r) 
ABC and ADE, give DE = Pe pes a 
gx? wed T? 
; hence 
ap? (e+r)? ‘ 
the base of the cone is ey, and its 
3 
solidity is Merny To facilitate the 
differentiation of this function, we divide 
by the constant 7-7 and reduce to the lowest terms; when our 
(e-+r)*. 
2-—r 
The first and second 
equation may be expressed, V= 
ee et eer RRR NEE an Sp Sone 6 
dr t-—?r dr? (x—r) 
Sxconp Serres, Vol. V, No. 14.—March, 1848. 30 
