42 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
From the associate formula we have 
B+C 
V=H 
2 
-*A); 
but by Newton’s formula 
V=|(B+C+4S*). 
Eliminating (B-f-C) we get 
Y=H(S 1 /s + t VA) = II(S 1 /2 +|T). (4) 
For the prismatoid it is readily seen that the perimeter of the mid-sec¬ 
tion is the half-sum of the perimeters of the bases; and the perimeter of 
base of the associate pyramid is the algebraic difference of perimeters of 
the bases. Note that it is necessary to regard all contra-clockwise motion 
as negative, and all lines thus generated. The perimeter of the mid¬ 
section of the associate pyramid is, therefore, half the difference of the 
perimeters of the bases. 
II. 
8 . To find an expression for the mean area of a prismoid in terms of the 
bases and another section. 
It is thought that the following method of deriving such a prismoidal 
formula is published for the first time. 
Let B, C be respectively the lower and upper bases of a prismoid. 
Let S be any section parallel to B and C. 
H=altitude of prismoid BC. 
a=altitude of prismoid BS. 
c=altitude of prismoid CS. 
A a =base of associate cone for prismoid BS. 
A c =base of associate cone for prismoid CS. 
A=base of associate cone for prismoid BC. 
M=mean area of prismoid. 
x=a-4-II. 
z=c^H. 
Yol. BS = 
B-f-S 
Yol. CS: 
)a ( B+S 
aS a! 
) l 2 
6H a y 
)e f° +S 
J C ~l 2 ~ 
6H sA J 
.•.By adding (5) and (6) we get the total volume. 
S(a+c) aB cC a 3 +c 3 
2 + T+ _ 2 -A “6FB r 
But V=MII .-. M=Y-hH 
x 3 +z 3 
. M _s. + ?? + yc 
. . 1V1— 2 ' 2 ' 2 
But M=^i?-^A. 
6 
( 5 ) 
( 6 ) 
t?) 
( 8 ) 
& 
