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PROFESSOR M. J. M. HILL OH THE LOCHS OF SINGULAR POINTS 
Art. page 
4. —Investigation of the conditions which are satisfied at any point on the Locus of Conic 
Nodes ......... . 178 
5. —Investigation of tlie conditions which, are satisfied at any point on the Locus of Biplanar 
Nodes. 179 
1.—Investigation of the conditions which* are satisfied at any point on the Locus of Uni- 
planar Nodes .. 181 
7. —Examination of the form of the Discriminant, and calculation of its Differential Coeffi¬ 
cients of the First and Second Orders... 182 
8. —Proof of the Envelope Property. 183 
9. —To prove that if E = 0 be the equation of the Envelope Locus, A contains E as a factor 
once and once only in general. 184 
Example 2. 185 
Example 3. 186 
10. —To prove that if C = 0 be the equation of the Conic Node Locus, A contains C 3 as a 
factor in general .. 186 
Example 4. 186 
11. —To prove that if B = 0 be the equation of the Biplanar Node Locus, A contains B 3 as a 
factor in general. 188 
Example 5 . . .* . . . . . ... . . . . . . . . . 189 
12. —To prove that if U = 0 be the equation of the Locus of Uniplanar Nodes, A contains U 6 
as a factor in general .. 191 
Example 6. 194 
Example 7. 196 
Section III. (Arts. 13-15 ).—Consideration of tlie cases reserved in the /previous section, in which two systems 
of values of the parameters satisfying the equations , D//Dn = 0, D//D6 = 0, coincide at a point on the 
Locus of Ultimate Intersections. 
13. —To prove that if each Surface of the System have Stationary Contact with the Envelope, 
then A contains E 3 as a factor. 202 
Example 8. 205 
14. —To prove that if the Conic Node Locus be also an Envelope, A contains C 3 as a factor . 207 
Example 9. 213 
15. —To prove that if the Edge of the Biplanar Node always touch the Biplanar Node Locus, 
then A contains B 4 as a factor .. 214 
Example 10. 218 
Section IV. (Arts. 16-25 ).—Consideration of cases reserved from the previous section. The degree of 
f (x, y, z, a, b ) in a, b is now the second , and the equations D//Da = 0, D//D5 = 0 are indeterminate 
equations for the parameters at points on the Locus of Ultimate Intersections. 
16. —The Discriminant and its Differential Coefficients as far as the third order. 223 
17. —The relations which hold good at points on the Locus of Ultimate Intersections . . . 226 
18. —To prove that under the conditions stated at the head of this Section, every Surface of 
the System touches the Locus of Ultimate Intersections along a Curve ..... 234 
