32 
TRANSACTIONS OP THE TEXAS ACADEMY OP SCIENCE. 
Consequently n 0 is eliminated whenever 
i— K x +y)+i x v=°- 
This beautiful criterion, graphically an equilateral hyperbola, is in 
marked contrast to the results given by Prof. W. H. Echols. 
Prof. Echols gives as his position-curve an ellipse 
II 3 — 1 5 
T ¥ 
a form in which of necessity one variable enters differently from the 
other, and therefore not a true criterion, since in reality the two sections 
enter indistinguishably. 
If for my variables their reciprocals be taken, then ^xy=^x-\-^y —1 
becomes 2 = 3a;-f-3y — Qxy, a less desirable form of the equation, though 
of course the same criterion and the same criterion curve, the equilateral 
hyperbola. I presented this criterion to the Texas Academy of Science 
on April 5, 1895. It has never been anticipated. 
Assuming the criterion, we have left for the volume 
V=a(%x 9 —ix 2 y)/(x—y)S &ix —a(%y a —±xy s )/(x—y)S illy . 
This, reduced by use of the criterion, gives at once 
V=a[!(3— y)/(x— y)+i]S a/ x+a[!(3— x )/{y— a)+£]S a/ y 
Of this most instructive and self-explanatory two-term prismoidal 
formula all others are merely special cases. 
Writing for x and y their reciprocals, it takes the form 
V=a[|(l— 2y)/(x— 2 /)]S a/x +a[|(l— 2x)/(y— z)]S a/y . 
From the criterion 
ixy=$x+%y— 1, 
and its formula for mean area, 
A = [i(3— y)( x —^)+i]Sa/x+[i(3— x)/(y — a3 )+i]Sa/y? 
we get for z the coefficient of S a/X , 
z=x*/(%x a —2x-\-2), 
and precisely the same form for the coefficient of the other section S a/y . 
This one relation gives all the coefficients, and is the equation to the 
coefficient curve. 
