50 
TRANSACTIONS OP THE TEXAS ACADEMY OF SCIENCE. 
The distance (h) apart of the sections Si and S 3 is variable. .•. For 
maximum and minimum values of h we have 
dh 36x—36x 2 —6 _ 
dx~~ (6x—3) 8 
For maximum value of h, x=-J. 
For minimum value x = ^(3±y 3 )- 
For maximum value x=infinity. 
For minimum value h=:-7==.5776. 
\A3 
For real sections of the prismoid h varies from .5776 to .6666. The 
coordinates of the vertices of the equilateral hyperbolas give the sections 
that are nearest together. The height (h) is the ordinate between line 
OD and curve 123 in Fig. 7. 
Substituting the values of x 1 , y 1 in (E) and reducing, we get 
6xy —3x—3y-|-2 = 0. 
, 1,1 
If we make x=— and y=— we get 
u j y a 
iuv=i(v+u) — 1. (K) 
This is the form of the condition that Dr. Halsted presented to the 
Texas Academy of Science (April 5, 1895), and which he christened the 
criterion formula. 
16. From (25) we have 
S=xC-fyB —xyA. 
Combine this with (9) by eliminating (A), and we get 
3x— 1 a S . 2 —3x 
M = B 
-C 
(33) 
6x J 1 6x(l —x) 1 “[6(1 — x) 
[This proof was given by Mr. E. P. Scliocli, an engineering student in 
the University of Texas, in May, 1895. Afterwards the writer found it 
in an article published by Prof. Echols in 1895.] 
17. Dr. Halsted in his Mensuration, p. 129, by using the equation to 
the running section S, where 
S=ax s +bx+c (34) 
has derived the following formula: 
V=MH=H^ 
_nf 2z —3 _ . 6(z— 1) , 3- 
_6(z— 1) S 2 z 2 
where the section S 2 is ~ the height from B. 
in 35. 
6 
B 
] 
(35) 
Make x=i and substitute 
Z 
M: 
3x—1 
;B + 
s 
2—3x 
c. 
(36) 
6(1 —x) ~6x(l —x~ 6b 
Then at any other section Si at a distance yH from B, we have 
6 0-y) S 1 ^ 6y v 
But M=-KC+B)~frA. (38) 
, 37 ) 
