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AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
If, however, the root corresponding to the negative sign be taken, i.e., 
Z, If 
then 
(£Zj + xXj - yYf (3£Z, + xX x - yY x ) 
Hence the factor £ 3 arises exclusively from the substitution in f of one system of 
values of the parameters satisfying both the equations D//Da = 0, D//D6 = 0. 
A similar demonstration shows that the solution 
will also give rise to the factor £ 3 . 
Now it will be shown presently that £ = 0 is the unode locus, 
on the unode locus, 
Y V 
A3 = 0 ±3 
Z 3 ’ z 3 
0 . 
Hence at any point 
Hence, at such a point there are two values of x — a and two of y — b which 
vanish. Hence two systems of values of a, b, satisfying both the equations Df/Da = 0, 
D//D6 = 0 become equal; viz., the two values of the parameter a become equal to 
the ^-coordinate of the point, and the two values of the parameter b become equal 
to the y-coordina.te of the point. 
(B.) The Locus of Uniplanar Nodes is £ = 0. 
To find the singular points, it is necessary to find values of x, y, z satisfying all the 
equations 
a {x — a) 3 + /3 (y — 6) 3 — 3 n (z — ax - f- byf = 0, 
a (x —• a) 2 + 2 na (z — ax + by) = 0, 
/3 (y — b ) 3 — 2 nb (z — ax + by) = 0, 
z — ax + by — 0. 
The only solutions of which are 
x — a, y = b, z = a 2 — b 2 . 
