AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 271 
From (201) and (202) 
zx 2 — {x + l) 2 = 0 .(203). 
A part of the locus of ultimate intersections is, therefore, given by (200) and (203). 
In this case x, y, z may be considered to be functions of a only. It will be noticed 
that if (200) be satisfied, A = 0. But A = 0 does not suffice to determine this part 
of the locus of ultimate intersections. 
(ii.) Next take the other solution of the second of equations (198), viz. :— 
— ax + by — 1 = 0 .(204). 
Combining this with the first of equations (198), 
2x + 1 = 0 .(205). 
Hence, by (204), 
y = (2 — a)/2b .(206). 
Therefore, by (199), 
2 = 1 .(207). 
Hence another portion of the locus of ultimate intersections is given by (205) and 
(207). 
In this case the coordinates of any point on the locus of ultimate intersections may 
be regarded as functions of the single parameter (2 — ci)/2b. 
It will be noticed that if (205) be satisfied, A = 0 ; but A = 0 is not sufficient to 
determine this part of the locus of ultimate intersections. 
II. The Fundamental Equations are satisfied by values ofi the coordinates which 
are independent of the parameters. 
In this case all the surfaces of the system pass through a finite number of fixed 
points, or a fixed curve. 
Example 19. 
Let the surfaces be 
^ ( x , y, *) + «<£ {%, y> z) + H ( x > y> z ) — °- 
(A.) To find the locus of ultimate intersections, it is necessary to satisfy at the 
same time the above, and 
9 ( x , y , z) = 0 , 
x ( x > y> z ) = °- 
Hence it is necessary to satisfy 
xh (x, y, 2 ) = 0, 6 (x, y , 2 ) = 0, x ( x > 2A z ) = 0 
