THOMAS U. TAYLOR-PRISMOIDAL FORMULAE. 
53 
We thus see that the Koppe, associate, and Echols formulae all belong 
to one group or family. 
The parabola QRL is the coefficient curve for A, where OD=l, 
RG= T ^, Q0 = DL=—and OE=DN=^(3—y/ 3 ). The line KH is the 
coefficient curve for (S1+S2) where OC = DH=:^. 
If z= + W 2 I, we get another neat formula where 
M=KSi+S 2 — A). (43) 
if z= ± we £ et 
M=i(Si+S,—T). (44) 
20. To find any section graphically. 
By equating equation (12) with that of Newton and simplifying, we 
get 
S X =B—x(3B-J-C— 4 S 14 ) -|-x 2 ( 2 B+ 2 C—4S^) (45) 
This equation represents a parabola, and any section can be found 
graphically. 
When x = 0, S=B. 
x=.50, S=Su« 
x=l, S=C. 
In Fig. 9 lay off OJ=B, FG = Si/ 2 , and DM=C; then describe a para¬ 
bola through these points. Any section at distance x'H from the base can 
thus be found by measuring the proper ordinate. 
For another proof of this, see Grunert’s Archiv, Vol. XXXIX. 
(The two parabolas in Fig. 9 have no connection whatever.) 
The area of ODMFJ where OD=H gives the volume of the prismoid. 
