GEORGE BRUCE HALSTED-TWO-TERM PRISMOIDAL FORMULAS. 25 
The sides of the mid section are the arithmetic means of the parallel 
sides of the bases. 
The size of M is in general not immediately dependent upon B t and 
B 3 , though it may be expressible by means of their sides and angles. 
Only in some special cases, as for example the prism, the pyramid and 
its frustrum, can M be expressed directly by B 5 and B 2 . 
On the other hand every section S parallel to these three can be made 
to depend upon B 1? B 2 , M, and its distances from these, as will now be 
shown. 
Consider first the plane trapezoid ABGH (Fig. 1). 
Let JK be its medial, that is the sect joining the 
mid points of the non-parallel sides AH and BG. 
Let PQ be any parallel to JIv. 
The distance of the base lines AB and GH from 
one another call h, the distance from AB to PQ call 
h 1? and that from QP to GH call h 2 . 
Draw the straight HRST parallel to BG. Call the 
area of ABGH=T; of JKGH=T l5 of GHQP=T 2 ; 
of BGHT=P; then KSGH = |P, and GHRP=Ph 2 /h. 
From the similarity of the triangles ATH, JSH, QRH follows imme¬ 
diately that (T—P): (T 2 — Ph 2 /h) = h 2 :li 8 , and (T t —|P):(T —Ph 2 /h) 
=ih2;h 8 . 
Eliminating Pgives an equation from which T a is easily determined; thus 
T 8 =Th 2 (h 2 -h 1 )/h 8 +T 1 4h 1 h 3 /h 2 . 
(!)• 
In this substituting h—h t for h 2 gives 
T a =T-(3T-4T 1 )h 1 /h+(2T-4T 1 )h 8 /h 8 . 
Hence T a , that is the area of the trapezoid GIIQP, is a linear function of 
the trapezoids ABGH and JKGII, and a quadratic function of the dis¬ 
tance hq of its base QP from AB. 
This settled, suppose a prismoid with base B t in the plane of the paper, 
and project it perpendicularly upon this plane; then the projections 
of all sections parallel to B x are equal to the sections themselves. Let 
ABCDEFGH be such a projection of a four-sided prismoid (Fig. 1) and 
call the areas ABCD=B 1 ; EFGH = B 2 ; JKLM=M; NOPQ=S. Call 
the distances from B x to B 2 , from B x to S, from B 2 to S respectively a, 
ffljl a 2‘ 
Then from (1) 
GHPQ=ABGH(h 2 — h 1 )h 2 /h s -f-JKGH 4h 1 h g /h s ; 
or since h, h l5 h 2 are respectively proportional to a, a t , a 2 ; 
GFIPQ=ABGH(a 2 — a 1 )a 2 /a 8 +JKGH 4a 1 a g /a 8 . . . (2j. 
Similar relations hold for the surfaces BCFG, CDEF, ADEII. 
