THOMAS U. TAYLOR-PRISMOIDAL FORMULAE. 
37 
But the areas abed and BC12 are described clockwise and contra-clock¬ 
wise, or vice versa. 
A=—4Si^. 
(c) In E—BCD we have 
V=£HB, 
By the associate formula 
v = H (^-iA) 
=h(|+ 0—iB) = |HB. 
The foregoing investigation is, perhaps, sufficient; but it will be well 
to follow out the investigation for the prismatoid as a whole. Let us as¬ 
sume that the mid-section of each primary solid of the prismatoid is as 
indicated in the second column of the following table: 
Solid. 
Mid-section. 
Base of asso¬ 
ciate pyramid. 
Upper base. 
Lower base. 
Volume. 
B—EFG 
d 
4d 
4d 
0 
fHd 
E—BCD 
e 
4e 
0 
4e 
f He 
E—ABD 
f 
4f 
0 
4f 
ffH 
GE—BC 
s 
—4 g 
0 
0 
£Hg 
FE—AB 
k 
—4k 
0 
0 
f Hk 
The base of the associate pyramid of the prismatoid is thus equal to 
the sum of the upper and lower bases of the prismatoid less the bases of 
associate pyramids of the tetrahedra that have only edges in the bases. 
It is only necessary to show that the generator of the associate pyramid 
of the prismatoid graphically sums up these individual bases. Figure 
2 shows the associate pyramid to the right of the prismatoid. The 
lower base is drawn complete for reference. By comparing the prisma¬ 
toid and the associate pyramid we see that 
Cl is equal and parallel to FE, 
21 is equal and parallel to BC, 
23 is equal and parallel to FG, 
34 is equal and parallel to AB, and 
4A is equal and parallel to GE. 
The parallelograms BC12 and B34A are the bases of associate pyramids 
of the tetrahedra FE—BC and GE—AB. If we subtract these parallelo¬ 
grams from ABCD and B23, there is left the base of the associate pyra¬ 
mid ADC1234A. 
From the last column of the table we have: 
V=y[8d+8e+8f+4g+4k], 
