AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 27-3 
For if z -j- c = 0 and a, b be finite, the equations are equivalent to 
x = 0, y = 0, 2c (c + cl) 2 = 0, 
which equations cannot be satisfied. 
Hence the values of a, b are infinite. 
IV. . The Fundamental Equations are equivalent to only two relations betiveen the 
coordinates and 'parameters. 
In such a case the discriminant must vanish identically. 
Example 21. 
Let the surfaces be 
oc(x — a ) 3 + 3/3 (y — b) 2 = 0, 
where a, (3 are fixed constants ; a, b the arbitrary parameters. 
The other fundamental equations are 
3 a (x — a) 2 = 0, 
6/3(y-b) = 0. 
Hence the discriminant vanishes identically. 
It may be noticed that in this case each surface of the system has a unodal line. 
Hence the singularity is of a higher order than when each surface has a single unode. 
V. The Fundamental Equations are equivalent to only one relation between the 
coordinates and qiarameters. 
In such a case the discriminant must vanish identically. 
Analytically 
[/(«, y, z, a, b)J = 0 
is an example. 
But the left-hand side is resoluble. 
MDCCCXCII.—A 
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