THOMAS U. TAYLOR—PRISMOIDAL FORMULAE. 
45 
x =i( 3 — V3), we get 
M=-.2886B+Si + .2886C. 
When x=i(3+^3), 
M=+.2886B-fS 8 —.2886C. 
By adding we get 
M=i(Si+S,). 
This can be readily obtained from the equation to the running section 
(S) where 
S=m-f-nx-f-tx 2 . 
H 
If x=-g (3— )fs), we have 
TITT fTJ 2 
S 1= =m+— (3— \^)+-gg (12—6V3). 
JJ 
If x=-w( 3+\/3), we get 
nH tTT 8 
S, = m+ T (3+V3)+^g(12+CV3). 
KS 1 +S.)H=H(m+^ ! + 5 f- ! )=V. 
11. By the use of equations (2) and (12) a two-term prismoidal form¬ 
ula can be derived by elementary geometry as follows: 
From (12) we have 
M = B 
3x—1 
S 5 
+C- 
2—3x 
6 x 1 6 x(l—x) 1 6 ( 1 —x)’ 
then for any other section (Si) distant yH from B we have 
3y—1 , S, , ^ 2—3y 
M = B- 
+ C; 
6 y 6 y(l—y) 6 ( 1 —y)* 
From (9) C = 2 M—B+a 
7 3 
By substituting this in (12) and (17) and reducing, we get 
M=B 
M=B 
2x—1 S 2 A(2—3x) 
2 x 
Si 
Eliminate B; 
M — Sj 
2 x 
2 y-i _ 
2 y ~ r 2 y 
2y—1 
6 
A(2—3y) 
) + & 1 2 (y-x) 
6xy—3y—3x-f2 
6 
A. 
(17) 
(18) 
(19) 
( 20 ) 
2 (y— 
Now if the coefficient of A in (20) disappears, we get a two-term pris¬ 
moidal formula. 
The condition for two-term formulae is, therefore, 
6 xy—3y—3x-f-2=0. (21) 
By transferring the origin to (|, |), we get 
x y=— tV (^) 
It is readily seen that this is an equilateral hyperbola referred to its 
asymptotes. 
