AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 223 
EiEf_ ( py y_ 0 
Da 3 D6 3 \DaDbJ 
holds, they are indeterminate. 
In this case the discriminant cannot be formed as in the previous section. 
There are not two coinciding systems of values of the parameters to consider. It is 
shown that there is one system which can be determined. 
There is also the additional peculiarity that the rationalising factor introduced to 
make the discriminant of the proper order and weight always vanishes at a point on 
the locus of ultimate intersections. Hence, on account of it, the equation of an 
envelope or singular point locus may be expected to enter into the discriminant one 
or more times. As this number cannot be determined in a general way, it is better 
to express the equation of the system of surfaces as a quadric function of the para¬ 
meters, and form the discriminant in the usual way. 
Art. 16.— The Discriminant and its Differential Coefficients as far as the third order. 
Let the equation of the system of surfaces be 
aa 2 + 2W ah -fi vh 2 -j- 2Va + 2U6 + w = 0 . . . . (142). 
To find the discriminant, solve for a, h, the equations 
wa + Wfc + V = 0.(143). 
Wa -f vh + U =■- 0.(144), 
obtaining hence 
a = 
b = 
wu - «v 1 
uv — W 2 I 
WV-tiU 
uv - 
(145). 
Now substitute these values of a, h in the left-hand side of (142). 
The result is 
The rationalising factor is 
Hence the discriminant 
u 
w 
V 
W 
V 
u 
Y 
u 
w 
u 
w 
W 
V 
* 
u W 
W v 
A = 
u W V 
W V U 
Y U 
u 
(146). 
