AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 235 
Now this, by (159), reduces to 
(X - x) | (uv - W 3 ) + (Y — y) | (uv - W 3 ) + (Z - z) | (uv - W 3 ) = 0, 
which is the equation of the tangent plane to the locus of ultimate intersections, since 
uv — W 2 = 0 at every point of the locus of ultimate intersections. 
Hence each surface of the system touches the locus of ultimate intersections along 
a curve. 
Art. 19.— To prove that under the conditions stated at the head of this Section, there 
are in general at every point of the Locus of Ultimate Intersections two Conic 
Nodes; and if C = 0 he the equation of the Locus of these Conic Nodes, A con¬ 
tains C 2 as a factor. 
(A.) To prove that there are in general two conic nodes it is necessary to show that 
there are in general two distinct sets of values of a, b, which satisfy (142), (161), 
(162), (163). 
These will be satisfied if (143), (161), (162), (163) be satisfied. 
Eliminating b from (143) and (161) the result is 
a 2 (W- 2wWW. r + v- x u % ) + 2a (iNv x - VWW, + W 2 Y, - WwU,) 
+ (Nv x - 2WVU, + W *w x ) = 0. 
Hence by (153), (154), (157), after division by u, it follows that 
a 2 1 (uv - W 2 ) + 2a|- (Nv - UW) + (vw - U 2 ) = 0 . . (180).'" 
And in like manner by eliminating a between the above equations 
6 2 | ( uv _ w 2 ) + 26 J^(U u ~ VW) + 9 - (uw - V 2 ) = 0 . . (181).t 
Further, by means of (159), it is possible in these equations to change x into y or 
into z. 
These equations will be called the parametric quadratics. 
Hence choosing a and b to satisfy (143) and (161), they will also satisfy (143) and 
(162), and (143) and (163), 
Hence it is possible in general to find two distinct systems of values of a and b 
which satisfy (142), (161), (162) and (163), at points on the locus of ultimate 
intersections. 
* The mean of the values of a satisfying (1£0) is the value of a given by (170). 
t The mean of the values of b satisfying (181) is the value of b given by (171). 
2 H 2 
