THOMAS D. TAYLOR—PRISMOIDAL FORMULAE. 
43 
Substitute the value of A found from (9) in (8) and we get 
M_ x—x 3 —z 3 ) _S_(;(y —x 3 — z 3 ) 
2 (l-x 3 -z 3 ) ' 2(1 —x 3 — z 3 ) 1 2(1 — x 3 —z 3 ) 
Now x-]-z=l x 3 +z 3 = l — 3xz(x+z)=l — 3xz 
M = 
B(3x—1) . S 
6 x 
6 xz 
3z—1 
6 z 
( 10 ) 
( 11 ) 
B(3x—1) 
S 
:+C 
2—3x 
6(l-x) 
( 12 ) 
6 x 6x(l —x) 
-fB+eS+dC, (13) 
where /, e, d represents the coefficient of B, S, and C, respectively. 
1 2—3x 
A d — e.( i_^14) 
n _3x—1 
*'• f_ 6x ’ e ' 
6 x(l —x)’ vl 6(1 —dx)* 
Transfer the origin for f to (0, %) and for d to (1, £) and we get 
( i5 ) 
which are the equations to equilateral hyperbolas referred to asymptotes 
OB and EF, and PQ and EF. 
If for e we transfer origin to (4, 0) we get 
2 
e= 
3— 12x 2 ’ 
(16) 
9. By making x vary from 0 to 1, we can get the following table of 
coefficients: 
X. 
f. 
e. 
d. 
0 
.05 
—2.833 
3.508 
.325 
.10 
—1.167 
1.350 
.315 
.15 
— .611 
1.307 
.304 
.20 
— .333 
1.042 
.292 
.2114 
— .2886 
1.00 
.2886 
.25 
— .1666 
.888 
.277 
.30 
— .0555 
.7936 
.262 
.1 
ft 
0 
.7500 
.250 
.35 
.00238 
.732 
.243 
.40 
.0833 
.6946 
.2222 
.45 
.129 
.673 
.1969 
.50 
.167 
.6666 
.1667 
.55 
.197 
.6730 
.130 
.60 
.222 
.6946 
.083 
.667 
.250 
.7500 
00 
.70 
.262 
.7936 
.0555 
By using x as abscissas and the corresponding coefficients as ordinates, 
we get the “coefficient” curves as shown in Fig. 6. It is noticed that 
the “coefficient” of S is symmetrical with reference to the line Kl, and 
