GEORGE BRUCE IIALSTED-TWO-TERM PRISMOIDAL FORMULAS. 
31 
III. 
By far the simplest and most valuable of all prismoidal formulas is that 
due to Kinklin, written in Halsted’s Geometry 
D=±a (B+3T), 
where D is an abbreviation for prismoid, a its altitude, B one base (either), 
and T the cross section at fa from the chosen base B. This two-term 
formula may be proved by elementary geometry very simply as follows: 
By planes through the edges project the 
prismoid perpendicularly on the lower base 
B. These planes cut the prismoid into one 
prism and a series of pyramids, some like 
ACD (Fig. 6) with three-sided base in B, 
the others all like ABC with four-sided base 
in the projecting-plane AB. This latter 
pyramid equals a prism with base ABC and 
altitude a, lacking a pyramid with base 
ABC and altitude a. Therefore, putting B 
for ABC, its volume is faB. Its cross section T is 
(ABC—f ABC) =fABC=fB. 
Consequently its volume is Ia(B+3T)=£a(B-|-tB) = 3aB. Likewise the 
volume of the pyramid ACD is given by the formula Ia(B-[-3T), as is 
also that of the prism. Therefore the algebraic sum of all these, that is 
the volume of the whole prismoid, is given by the formula D=Ia(B-j-3T). 
IV. 
The criterion for two-term prismoid formulas is 
$xy=%x+$y-l. 
Proof. In the formula alread}' - established by elementary considera¬ 
tions in Section I of this paper for the area of any cross-section, 
A x=/(a) =n 0 + WjSB+n.a! 8 , 
let a; be l/y part of the altitude, then 
s a/y = ^ 0 + n i a /2/+ ? C a V2/ 2 - Symmetrically 
Sa/x = * 0 + w » a /®+ w « a Y**- 
Hence w i = [® S Sa/x— (^ ~ V 2 ) n 0 — y* S a/y]/(® — y) a - 
n 2 = Oy 2 S a/y + (x 2 y- xy 2 ) n 0 - x 2 yS a , x y (x—y) a 8 . 
Therefore the volume 
y=aw 0 +|a 2 w 1 +^a 3 R 3 , becomes 
V= an 0 +\ax 2 S a/X / (x—y)— \an 0 (®+y) - \ay 2 S a/y / (x-y) + 
laxy 2 S a/y / (x—y) + ian 0 xty—iax 2 yS. d / x / (x—y ). 
