THOMAS U. TAYLOR-PRISMOIDAL FORMULAE. 
49 
Now we wish to find the mean area of the section P between sections 
C and B in terms of S 1? S 3 and A h- Using the well known formula for 
mean value for y=F (x) between the limits of a and b for x we have 
M= mean value=- f a F(x)dx ^27) 
3 .—by b 
From (26) then 
M= a^0 ( H h r +a) A ] d z 
_H—2a s 2h—H+2a s _3hH—2H 2 +6aH—6ah—6a 2 A 
2 h 1 2 h 2 6IP A 
2 h 1 1 2 h 
Now make h=c—a where c=distance B. 
H—2ag j_2c—H q 3Hc-J-3aH—6ac—2H 
. \ M= 
_-S„—- 
2(c—a) 3 
6H S 
2 A 
(28) 
(29) 
2 (c—a) 
Make a=xH and c=yH. 
M- 2y-l s + + 6xy-3x-3y+2 A 
2(y—x) 2(y—x) 6 
Now, as in (20), for a two-term formula, the coefficient of A must dis¬ 
appear: 
.*. 6xy—3x—3y-|-2=0 (30) 
andxy=—(31) 
1—2x 2y—1 
P=— —> _< » (32) 
2(y—x) 2(y—x) 
15a. Prof. Echols gave as the condition of a two-term formula 
x' 8 +x'y'-fy' 2 — f(x / +y') = 0. 
Where x'h=a, and y'h=H—x'h. 
If we impose the condition that the sections (for convenience) are be¬ 
tween bases, this formula becomes 
x' 2 — x'y'+y' 2 — f(y'— x')= 0 . (E) 
But H=h(y'+,x') 
a=x'h=xll 
y'h=H—xH=H(l—x) 
h=yH-xH=H(y-x)=_ H 
y'+x' 
X =- 
and y' = 
y—x 
1-X 
y—x 
l 
x'+y'=- 
y—x 
From Equation (21) we have 
3x—2 
6 x — 3. 
But h=(y—h)H 
fix—6x s —2 
6 x—3 
H. 
