44 
TRANSACTIONS OP THE TEXAS ACADEMY OF SCIENCE. 
that the curves (f) and (d) are symmetrical, with respect to each other, 
with reference to the same line. 
The (e) curve is asymptotic to lines x=0(OB) and x=l(PQ). The 
(f) curve is asymptotic to OB and line a=^(FE). 
It can be seen from equation (16) that the (e) curve is asymptotic to 
lines x= 
10 . Coefficient curves for three-term formulae. 
From,Fig. 6, where OP=unity, we readily see that by taking values 
of x from 0 to 1 the ordinate to (d) curve gives the coefficients of C, the 
ordinate to (e) curve gives the coefficients of the floating section S, and the 
ordinate to the (f) curve gives the coefficients of B. When these sections 
are multiplied by their coefficients and added, a mean area is obtained, 
which, multiplied by the height of the prismoid, will give the true 
volume. 
When x = ^, the coefficient of B disappears, and we get 
M=4c+fS, 
thus verifying by the three-term formula Kinklin’s two-term formula. 
When x=-|, we get 
m=*b+4S, 
another verification of the same formula. This two-term formula can be 
considered a special case of three-term formula. By inspecting Fig. 6 or 
the table I, we see that if we wish positive coefficients we must take S in 
the middle third of the height, and, as it leaves the middle third , it verifies 
the two-term formula. 
The coefficients are symmetrical with reference to cd or the midsection. 
When x=-J, we get 
M = i(B+C+4S*). 
