48 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
(5) The ‘‘coefficient” curve crosses asymptote 0 1 A 1 , on an ordinate 
through vertex of “position” curve. 
(6) For the last position, S s is at a distance=.2114H from one base 
and Si the same distance from the other base, and the coefficients here 
are each 
(7) When x=0, y=.667, and the coefficients are .250 and .750 re¬ 
spectively, thus again verifying Kinklin’s formula. 
(8) When x=.333+i y=l, and the coefficients are .750 and .250 re¬ 
spectively. 
(9) At the points x=0, .333, .667, or 100, the “position” curve 
leaves the square, and the branches 728 and 456 give all the sections be¬ 
tween the bases. 
(10) The diagram and table show very plainly that for any real sec¬ 
tion (between bases) no section can occur in the middle third. The three- 
term formula for positive coefficients requires the section to be taken in 
the middle third. 
(11) The “coefficient” curve lgmn is plotted by making QO=unity. 
(12) The “ position ” curve is plotted by making OQ=OR=heigkt of 
prismoid. 
15. In the “Annals of Mathematics” for November, 1894, Prof. 
Echols has shown that there is an infinite number of two-term formulas 
that will give the mean area of the prismoid. It is the object of the 
present note to follow his general method in order to verify the calcula¬ 
tions of article 11. 
For convenience the two parallel sections will be taken between the 
bases C and B. 
to B, S 3 . Then, if P be any section, 
And the distance from P to section S 0 =m 
The section next to C will be called S x , and that next 
Distance between S x and S 2 =h 
And the distance from P to section S 1 =h—m 
Distance between section B and S 2 =a 
Distance between bases B and C=H 
Distance between P and base B=z 
From Elliott’s extension of Holditch’s theorem we have 
+ h-m m(h-m) 
h 1 h 2 h a 
(25) 
Where A h = area of base of “associate cone” between sections Sj and 
It is clear that 
• P -Tr s ‘+- 
-z-j-a 
S 3~ 
m=z—a 
(z—a) (h—z-(-a) 
Ah 
(26) 
