GEORGE BRUCE HALSTED-TWO-TERM PRISMOIDAL FORMULAS. 
21 
plies exactly to all solids whose cross-sections are cubic functions of the 
section-height: 
If A x =f(x)=n 0 -{-n l x-{-in 2 x 2 -\-n 3 x 3 , 
then /(o)+4/( $«)+/( a) =n 0 
+4n 0 +2an 1 +ci 2 n 2 +%a 3 n 3 
-f- w 0 -f- an 1 -\-a 2 n„-\- a 3 n 3 
= 6n 0 -{-3ah 1 -\-2a 2 n 2 -\-%a 3 n 3 . 
Thus D = ia[B 1 +4M+B 3 ] = ia[/(o)+4/(^a)+/(a)] 
= ia[6n 0 +3an 1 +2a 2 n 2 4-fa 3 n 3 ]=:an 0 +^a 2 7i 1 -{-\a 3 n 2 -{-\a i n 3 . 
But by the calculus this is the exact volume of the solid, since it is 
This investigation is faulty, and does not fix the limit of applicability, 
since it says nothing to show that the conditions are satisfied only by 
functions which have no fourth or higher powers. This is proved in my 
Mensuration without the calculus. For readers of the calculus the fol¬ 
lowing method may be of interest: 
Measuring a: on a line normal to which the sections are made, let 
A x =/(cc) be the area of the section at the distance x from the origin. 
Let three sections be made through any solid at the distance ( x — h), the 
distance x, and the distance ( x-\-h) from the origin. Then f(x — h), 
f (%), f (x-\-h) will be the areas of these sections, and the old Prismoidal 
Formula, for the volume between the bases f(x — h) and f(x-\-li) gives 
^/i[/(x-7i)+4/(x)+/(a;+A)]. 
But the volume is the integral of the differential solid f(x)dx between 
the limits x — h and x-\-h. 
J‘ x+i ^f(x)dx=-Jf(x-{-h)dx—Jf(x—h)dx. 
If the function / fulfills the conditions of the Prismoidal Formula, we 
have, by equating the two expressions for the volume, 
ff(x+h)dx—ff(x—h)dx=$h[f(x—h)+4rf(x)+f(x+h)]. 
To find what form of / will satisfy the equation, develop both its mem¬ 
bers by Taylor’s Theorem. 
The first member becomes (A) 
/[/(*)+/' (*)»+/" (*)y+/'' >)^3+etc.]<& 
' '(ao^+etc.]* 
(x'jhtlx^ if ' '(.ej/'to+etc. ] 
=2 f(x)h+f ' (»)£+/ "' ■ 
