40 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
Prof. Echols in the “Annals of Mathematics” for November, 1894, 
says: 
“ The expression for the mean area 
M=i(B 1 + B 2 )-|A, (2) 
wherein A is the area of the base of the associate cone, is, I believe, due 
to Koppe, and seems to be the first prismoid formula found. This 
formula, in connection with 
(1) [M=i(B 1 + B s +4S■*)], 
easily furnishes the value 
M-Sa+^A. (3) 
In this form the mean area has been used by engineers for computing 
earthwork volumes. The writer’s first acquaintance with it thus em¬ 
ployed was in the lectures to engineering students at the University of 
Virginia in 1880. This is a two-term formula, but it involves an extrane¬ 
ous area, that of the director cone, which is not a section of the solid.” 
Prof. Echols here makes the mistake of ascribing the wrong formula 
to Koppe. Interchange (2) and (3) and the remarks hold good. 
6a. If P ( vertex of associate cone ichose base is A) is in base C, then C 
is base of associate cone whose upper base is A. 
In Fig. 4 the upper base (C) is projected orthogonally on lower base 
B. The generator of the cylindroid is BC, while PA is generator of 
associate cone whose base is A. As BCPA is a parallelogram, we can 
conceive another cylindroid generated at same time by BA and its asso¬ 
ciate cone generated by PC. Then if A is projected orthogonally on 
upper base, there are two cylindroids generated simultaneously that bear 
a reciprocal relation to each other. 
If M=mean area of cylindroid whose bases are B and C, 
and M'=mean area of cylindroid whose bases are B and A, 
we have M=A(B-j-C)—4A, 
M'=KB+A)-*c. 
.-. M—M' = |(C—A), 
M+M'=B+KC+A). 
