GEORGE BRUCE HALSTED-TWO-TERM PRISMOIDAL FORMULAS. 
23 
For the second position 
x 2 B s 
: a 3 
For the third position, if the mid section —^na 3 , 
A x =n{ax—x 2 ) 
(see Halsted’s Geometry, page 250, whence are taken figures 4 and 5, 
and where was first given the Theorem: If a sphere be tangent to the 
parallel planes containing opposite edges of a tetrahedron, and the sec¬ 
tions made in the globe and tetrahedron by one plane parallel to these 
are equivalent, sections made by any parallel plane are equivalent). 
Fig. 5. 
But from the mid section a 2 of such a tetrahedron we can obtain 
the mid section of any tetrahedron in this position by three consecutive 
changes, each of which is equivalent to multiplication by a number. 
Representing the product of these three numbers by n, the desired mid 
sectionand any section A x = mr(ax —or). 
Thus in any prismatoid any cross-section is only a quadratic function 
of its distance from either base. Therefore in employing for its volume 
the old three-term formula, we have been using a bear-trap to catch a 
mouse. 
For all solids whose section is a function of degree not higher than the 
second, or 
/ 3 / 
f(x) =n 0 a-\-\n 1 a 2J r ^n 2 a' i . 
Measuring x from one base B 1? we have 
A 0 = B^ — n 0 . 
Then A a =B 3 = B 1 +w 1 a+w 3 a 2 . 
We see at once that any cross-section whatever, if known in addition 
to the altitude and bases, will give us the volume. 
