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PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
Table of Contents. 
Art. page 
Introduction. 141 
Part I.—The equation of the System of Surfaces is a rational integral function of the 
COORDINATES AND ONE ARBITRARY PARAMETER. 
Section I. (Arts. 1-6 ).—The factors of the Discriminant which in general correspond to Envelope and 
Singular Line Loci. 
1. —To show that if E = 0 be the equation of the Envelope Locus, the Discriminant contains 
E as a factor. 142 
Example 1. 144 
2. —To prove that the Locus of Conic Nodes of the Surfaces f (x, y, z, a) — 0 is a Curve, not 
a Surface. 145 
3. —To find the conditions which hold at every point on a Surface Locus of Binodal Lines . 147 
4. —To find the conditions which hold at every point on a Surface Locus of Unodal Lines . 148 
5. —Examination of the Discriminant A, and its Differential Coefficients when a Surface 
Locus of Binodal Lines exists. Proof that A contains B 2 as a factor. 149 
Example 2. 149 
6. —Examination of the Discriminant A, and its Differential Coefficients when a Surface 
Locus of Unodal Lines exists. Proof that A contains U 3 as a factor. 150 
Example 3. 152 
Section II. (Arts. 7-9 ).—Consideration of the cases reserved in which turn roots of the equation D//Da = 0 
become equal at any point on the Locus of Ultimate Intersections. 
7. —Consideration of the exceptional case of the Envelope Locus, in which two consecutive 
characteristics coincide. 153 
Example 4. 154 
8. —Consideration of Loci of Binodal Lines which are also Envelopes. 155 
Example 5. 162 
9. — Consideration of Loci of Unodal Lines which are also Envelopes. 165 
Example 6. 166 
Section III. (Arts. 10-11 ).—Supplementary Remarks. 
10. —Further remark on the case in which D 2 / 1 /Da 1 2 = 0 167 
11. —If the Surface f (x, y, z) = 0 have upon it a Curve at every point of which there is a 
Conic Node, then the tangent cones at the Conic Nodes must break up into two 
planes. 168 
