46 
TRANSACTIONS OP THE TEXAS ACADEMY OF SCIENCE. 
The coefficients of S 3 and Si are 
2y- 1 _ 1 i_2x 3(t-2x) 8 _ 
p 2(y—x) 4(3x 2 —3x-)-l)’ q 2(y—x) 4(3x 2 —3x+l)’ ' ' 
Transfer the origin to (+ 0) 
1 12x 2 .... 
' * P_ l + 12x 2 ’ q ~~l-f-12x 2 ’ 
If OR=OQ=H, say 100, and OA and OC represent the respective 
axes, the branches of the hyperbola can be plotted, and the ordinates and 
abscissas to the branch 123 between 7 and 8 and 456 between 4 and 6 will 
locate all the sections between the bases. (Fig. 7.) 
The “coefficient” curve is obtained from equations (23), where for 
any value of x, the ordinates from OA to the coefficient curve will give 
the coefficient of S 2 and the ordinates dozen from QD to the same point on 
the coefficient curve will give the coefficient of Si. 
12 . easy calculation the following values of x, y , p, and q can be 
found: 
X. 
y . 
Coeffi 
Of Si. 
q- 
cients 
Of So. 
P - 
h. 
.00 
.666 + 
.750 
.250 
.6667 
.05 
.685 
.709 
.291 
.635 
.10 
.709 
.659 
.443 
.609 
.15 
.738 
.595 
.405 
.5880 
.20 
.778 
.519 
.481 
.5780 
.2114 
.7886 
-.500 
.500 
.5776 
.25 
.833 
.429 
.571 
.5833 
.30 
.917 
.324 
.676 
.617 
.331 
1.000 
.250 
.750 
.6667 
.35 
1.056 
.212 
.788 
.756 
•371 
1.16* 
.158 
.842 
.7916 
.40 
1.333 
.108 
.892 
.9333 
.43 
1.693 
.0556 
.9444 
1.063 
.45 
2.167 
.029 
.971 
1.717 
.47 
3.278 
.011 
.989 
2.808 
.48 
4.66§ 
.00481 
.995 
4.186 
.49 
8.833 
.002 
.998 
8.343 
.50 
Infinity 
0 
1.000 
Infinity 
.60 
— .33^ 
.108 
.892 
.667 
.0 
.250 
.750 
.70 
.083 
.324 
.676 
.80 
.222 
.519 
.481 
.90 
.292 
.657 
.343 
1.00 
• .333 
.750 
.250 
— .10 
.639 
.812 
.188 
— .20 
.619 
.854 
.146 
