THOMAS U. TAYLOR-PRISMOIDAL FORMULAE. 
39 
Differenzen der gleichliegenden Seiten der beiden Grundfliichen des Obe- 
lisken gleich sind.” 
It is thus seen that Koppe’s theorem as stated here is confined to the 
prismoid (obelisk); that is, to solids whose bases are equiangular. This 
theorem of Koppe was easily established by elementery geometry for the 
old prismoid. The bases of the prism and pyramid were polygons whose 
perimeters were respectively the half-sum and the half-difference of the 
perimeters of the bases, and whose angles were equal to the correspond¬ 
ing angles of the bases. 
On page 20 of his “ Neuer Lehrsatz” Koppe remarks: “The appli¬ 
cation of our proposition to bodies that are bounded (enclosed) by 
curved (or crooked) side faces does not confine itself to the frustrum of 
a cone, but extends to all bodies which have two parallel bases and 
curved (or warped) side-faces, that are generated by a straight line 
which is alwa} r s on the curved lines that circumscribe the bases.” 
6. Koppe’s formula can be established for the prismatoid by the same 
methods we employed for the associate formula. We must bear in mind 
that the perimeters and areas of sections of the associate pyramid when 
described contra to those of the bases must be treated negatively. 
Refer to Figs. 1, 2, and 3, and table in Art. 3. 
(a) B—EFG. 
We know that 
V=^HC. 
By Koppe’s formula 
V-H(S*+*T)=H(d+*d)«fHd=*HC. 
(b) 
BC-EF. 
V=|H g. 
By Koppe 
V=H(Si /2 +fT)=H( 
(c) 
E-BCD, 
V=^HB. 
By Koppe 
V=H(S*H4T) = H(f+*f)=$Hf=*HB. 
The application of the formula can be extended to the prismatoid as 
' a whole. 
From the last column of table have 
y=|-(8d+8e+8f+4g+4k. 
=^[(6d+6e+6f+6g-l-6 k )+( 2 d-f2 e + 2f —2g— 2k)]. 
=|[(6S 1/2 +2T)=H(S 1/2 +iT). 
Its extension to the cylindroid follows, as in Art. 3. 
