22 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
The second member becomes (B) 
^h[f(x)-f(x)h+f' : (x)j-eU:.+4f(x)+f(x)+f'(x)h+f'’(x)h‘+etc.] 
= ih[6f(x)+f"(;x)h‘+f"(x)^ i + e tc. 
= 2/<*)»+/"(*)£+/''" W^+etc. 
Comparing the last members of (A) and (B), we find the equation, 
*/(»)*+/" (*)£+/’''' (»)j£+eto.= 
= 2/(x) h+f ■• (*)f +/'' " (*) U+etc. 
Therefore the old Prismoidal Formula applies exactly to all solids con¬ 
tained between two parallel planes, of which the area of any section par¬ 
allel to these planes can be expressed by a rational integral algebraic 
function, of a degree not higher than the third, of its distance from 
either of these bounding planes or bases. And in general it applies uni¬ 
versally to no other solids. 
Thus the cubic 
A x =-r 0 -\-n x x-\-n. 2 x 2 -\-n^x 3 
expresses the law of variation in magnitude of the plane generatrix of 
old prismoidal spaces. 
But our prismatoid needs only a quadratic. This is readily proved. 
Any prismatoid may be divided into tetrahedra, all of the same altitude 
as the prismatoid; some having their apex in the upper base of the pris¬ 
matoid, and for base a portion of its lower base; some having base in the 
upper, and apex in the lower base of the prismatoid; and others having 
for a pair of opposite edges a sect in the plane of each base of the pris¬ 
matoid. A section A x of a tetrahedron in the first position equals 
(a —cc) 8 B 1 . 
a 
