GEORGE BRUCE HALSTED-TWO-TERM PRISMOIDAL FORMULAS. 
29 
portant paper. Whatever he may have obtained from Becker’s “Com- 
binatorik,” of which I had never before heard, it is certain that Sinram 
obtained geometrically a general formula for the prismoid in terms of 
two cross sections, thus demonstrating by elementary mathematics in 
1879 the existence of an unlimited number of such two-term formulae, 
involving sections of prismoid only. 
This result was published fifteen years later by Professor W. H. Echols 
of the University of Virginia in the Annals of Mathematics, Vol. IX, 
No. 1, in an article “On the mean-area of the Prismoid.” But Sinram 
had also investigated the mean area and indicated even a one-term pris- 
moidal formula, using a true mean area, only for this as he says, the 
divisor in the one-term formula would be irrational. 
Since Sinram reached all these results geometrically, this part of the 
subject was exhausted. 
II. 
Leaving for a moment these innumerable true two-term prismoidal 
formulas, where the two sections are really cross sections of the prismoid 
itself, mention may be made of a pseudo two-term prismoidal formula, 
which uses an extraneous area, that of the so-called “ associate-cone ”- 
base, which is not a section of the prismoid. 
If a straight passing always through one same point in the plane of 
one base of a prismoid moves so as to be always parallel to the straight 
generating the prismoid, it cuts out a cone between the basal planes. 
This is named the “ associate cone.” Calling the area of its base B c , the 
volume of the prismoid 
V=a(M+*B c ). 
This, in a paper in Scientiae Baccalaureus, Feb., 1891, I correctly at¬ 
tributed to Koppe (Crelle’s Journal, Vol. 18, p. 275; afterwards in his 
“ Ein neuer Lehrsatz der Stereometrie,” by Karl Koppe, 1843, §5, p. 5.) 
But in that paper I deduced it as an immediate corollary from a theo¬ 
rem there proved given in my Mensuration (1881) as follows: Twice the 
volume of the segment of a ruled surface between parallel planes is equiv¬ 
alent to the sum of the cylinders on its bases, diminished by the cone 
whose vertex is in one of the parallel planes and base in the other, and 
whose elements are respectively parallel to the lines of the ruled surface. 
This gives for the “ mean-area” of the prismoid (that area whose pro¬ 
duct by the altitude is the volume) the expression 
A=KB 1 +B 8 )—iB c ; or: 
The mean area of a prismoid equals the half-sum of the bases, less one- 
sixth the base of the associate cone. 
This theorem which Professor Taylor has since called “ the Associate 
