28 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
“ For all solids whose section is a function of degree not higher than 
the second, or A x =qfmx-\-nx B , g, m, n, and consequently A x for all values 
of x, are determined if the value of A x for three values of x is known. 
“Measuring x from B, we have A =B =g. 
“ Supposing we know the section at \/z the height of the solid above 
B x , we have for determining m and n the two equations, 
B 2 = B x +ma-f-na 8 , 
Aa = B ^ma/z+na* /z B . 
Z 
“ For the volume of the solid we have, by Cor. 2, page 123, 
V—B 1 a-f^ma B -\-^na 3 , 
or V=[a/6(z— l)][(2z—3)B 2 — (z —1) (z —3)B 1 +2 8 Aa ]. 
Z 
“ For 2=3, this gives 
Y==ia(B 8 +3Aa). 
3 
“Again, for z= 1-J, 
V=ia(Bi+3 A% a ). 
“These give the following theorem: 100. To find the volume of a 
prismatoid, or of any solid whose section gives a quadratic: 
“ Rule: Multiply one-fourtli its altitude by the sum of one base and three 
times a section distant from that base two-thirds the altitude.” 
On the last sheet of the MS. of Kinklin’s “Theory of the Prismoid,” 
lent me by Prof. T. U. Taylor, I find the following in continuation of 
the MS. in the handwriting of the copyist: 
“Grunert’s Archiv, Vol. LXII, 1879, pp. 440-3. 
“new evaluation of the volume of a prismatoid. 
“ Th. Sinram. 
“Since long it has been customary in evaluating the Prismatoid, as 
also the Obelisk, to use both bases, the mid cross-section, and the altitude 
of the solid, according to the formula 
Yol.=-iA[G+<7+4D]. 
“ In calculating these bodies the thought was always with me (J. K. 
Becker has handled this question in his ‘ Combinatorik’), whether a form¬ 
ula could be deduced geometrically in which should appear either only two 
cross-sections, or, indeed, only one. Then the general formula would be 
Vol.=D7i/a. 
“ I have attained the first, yet not the last, since a will be irrational, 
whereby” 
The sheet of MS. breaks off thus in the midst of a sentence, and this is 
all I have seen, but is enough to clearly indicate the content of this im- 
