264 
PROFESSOR M. J. M. HILL ON THE LOCHS OF SINGULAR POINTS 
(D.) If* = 0, A appears to vanish, but then the equation of the Surfaces, 
3/3 (x - a)H + cz s + 3d (y — b)~ -fi ez~ = 0, 
45 of the Second Degree in the parameters, and f the Discriminant be formed it does 
not really vanish. 
For the discriminant required is not that of the cubic 
3/3tX 2 Z + 3dY : Z + (cz 3 + ez*) 
but of the quadric 
3/3zX 2 -b 3dY' 2 -f (cz 3 + ez~) Z\ 
It is therefore 
9/5dz 3 (cz J r e). 
(E.) The Locus of Biplanar Nodes is now z— 0, the edge of the Biplanar Node 
being in the Biplanar Node Locus. 
The edge satisfies the condition for contact with the biplanar node locus. Hence 
the factor 2 3 is accounted for (Art. 21). 
(F.) If e — 0 , the Locus of Uniplanar Nodes is 2=0. 
In this case the discriminant is 9f3cdz 4 . Hence the factor 2 4 is accounted for 
(Art. 22). 
Section Y. (Arts. 26-29).— The Intersections oe Consecutive Surfaces. 
It has been shown that when the analytical condition (76) is satisfied which expresses 
that the fundamental equations are satisfied by two coinciding systems of values, the 
number of factors in the discriminant corresponding to conic node, biplanar node, and 
uniplanar node loci, is less when the degree of the equation in the parameters is the 
second than when it is of a higher degree. 
It has also been shown that, when (76) holds and the degree in the parameters is 
the second, each surface of the system, its consecutive surfaces, and the locus of 
ultimate intersections, intersect in a common curve. 
It is desirable, therefore, to examine the nature of the intersections of consecutive 
surfaces in all other cases. 
Art. 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 contact with each Surface of the System, and then there are tzvo points 
of intersection. 
(A.) First consider the case of an ordinary envelope. 
Let ch rj, l be a point of intersection of the surfaces 
