AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 277 
Art. page 
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 be the equation of the Locus of these Conic Nodes, A contains C 3 as a factor . 235 
Example 11 ...... 236 
20. —To prove that under the conditions stated at the head of this Section, if the two Surfaces 
having Conic Nodes coincide, then they are replaced by a single Surface having a 
Biplanar or a Uniplanar Node. 243 
21. —If the two Conic Nodes are replaced by a single Biplanar Node, and if B = 0 be the 
equation of the Biplanar Node Locus, and if the Edge of the Biplanar Node touch 
the Biplanar Node Locus, A contains B 3 as a factor .. 244 
Example 12. 244 
22. —If the two Conic Nodes are replaced by a single Uniplanar Node, and if U = 0 be the 
equation of tlie Uniplanar Node Locus, A contains U 4 as a factor. 248 
Example 13. 250 
23. —If the parameters of one of the two Surfaces having Conic Nodes become infinite, and 
if C = 0 be the equation of the Conic Node Locus, A contains C 3 as a factor ... 253 
Example 14. 253 
24. —If the parameters of both of the Surfaces having Conic Nodes become infinite, and if 
E = 0 be the equation of the Envelope Locus, then A contains E 3 as a factor . . . 255 
Example 15. 258 
25. —If the parameters of both of the Surfaces having Conic Nodes become indeterminate, 
then at evei’y point of the Locus of Ultimate Intersections there are an infinite 
number of Biplanar Nodes; each Surface of the System has a Binodal Line lying on 
the Locus of Ultimate Intersections; and if the locus of these Binodal Lines be 
B = 0, then A contains B 4 as a factor. 259 
Example 16. 262 
Example 17. 263 
Section V. (Arts. 26-29 ).—The Intersections oj Consecutive Surfaces. 
26. —To prove that the Surfaces represented by the three fundamental equations intersect in 
one point on the Envelope Locus, unless the Envelope Locus have Stationary Con¬ 
tact with each Surface of the System, and then there are two points of intersection . 264 
27. —To prove that the Surfaces represented by the three fundamental equations intersect in 
two points on the Conic Node Locus, unless it be also an Envelope Locus, and then 
there are three points of intersection. 266 
28. —To prove that the Surfaces represented by the three fundamental equations intersect in 
three points on the Biplanar Node Locus, unless the Edge of the Biplanar Node 
always touch the Biplanar Node Locus, and then there are four points of intersection 268 
29. —To prove that the Surfaces represented by the three fundamental equations intersect in 
six points on the Uniplanar Node Locus. 269 
