38 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
H 
=-g-[12d+12e+12f—(4g+4e+4f—4g—4k)], 
H 
=-g[3C+3B—(4d+4e+4f—4g—4k)]. 
But from third column 
A=4d+4e+4f—4g—4k, 
II 
V==-g(3C+3B—A) 
-h( c+b 
2 i A )* 
( 2 ) 
I have called this formula the “ associate” 
name. I do not know who first published it. 
formula for lack of a better 
It was published as prob¬ 
lem 440 in Halsted’s Mensuration in 1881. It was given at the Univer¬ 
sity of Virginia by Prof. Thornton as early as 1880, at least. I have 
never seen a proof of it, except for the old prismoid. 
If in the bases of the cylindroid we inscribe polygons, we can by join¬ 
ing the vertices of the different polygons with each other form a prisma- 
toid. If the number of sides of these polygons be indefinitely increased, 
the bases of the prismatoids gradually approach the bases of the cylin¬ 
droid as a limit, and the base of the associate pyramid approaches the 
base of the associate cone. 
If V'=volume of prismatoid, 
M'=mean area of prismatoid, 
V=volume of cylindroid, 
M=mean area of cylindroid, 
then the ratio of V' to M' is always constant and equal to H, there¬ 
fore their limits V and M have the same ratio. 
V' V 
= Y7 = II. 
M' — M 
limit of 
B'+C' 
-iA > 
B+C 
_i A 
5. KOPPE’S THEOREM. 
In 1838 Karl Koppe, Oberlelirer am Gj^mnasium zu Soest, published 
in Crelle’s Journal the theorem that now bears his name. In 1843 it was 
republished in his “ Ein Neuer Lehrsatz der Stereometrie,” a pamphlet 
that was intended-to be a supplement to all text-books on Stereometry. 
The theorem as enunciated by Koppe was as follows: 
“ Yeder Obelisk ist gleich der Summe aus einem Prisma und einer Pyr- 
amide, welche beide mit dem Obelisken gleiche Hohe haben, und deren 
Grundflachen in den Winkeln mit den Grundfliichen des Obelisken 
iibereinstimmen, walirend die Seiten der Grundflache des Prismas den 
halben Summen und die Seiten der Grundflache der Pyramide den halben 
