266 PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
by [ 1 , (184) by — [a, /3], (185) by [a, a], and adding, to form a new equation in which 
the lowest terms in X, Y, Z are of the second degree. 
Hence the equations (183)—(185) are equivalent to three others in which the lowest 
terms in X, Y, Z are of degrees 2, 1, 1 respectively. Hence there are two sets of zero 
values of X, Y, Z. Hence there are two intersections. 
Art. 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. 
(A.) In the case of the Conic Node Locus [£] = 0, [ry] = 0, [£] = 0. 
Hence the lowest terms in X, Y, Z in (183) are of the second degree, in (184) and 
(185) of the first degree. 
Hence there are two intersections. 
(B.) In the case where the conic node locus is also an envelope, it will be shown 
that the values of X, Y, Z, which make 
[«, f] X + [a, y] Y + [a, £] Z = 0.(188), 
[(3, fl X + [A y] Y + [A 0 Z = 0.(189), 
also make 
[£ a X 2 + [y, y ] Y a + B, Q Z ? + 2 [y, Q YZ 4- 2 B, f] ZX + 2 [£ y] XY = 0 (190), 
so that the lowest terms in the equations, by which (183)—(185) may be replaced, 
are of degree 3, 1, 1 respectively, and hence there are three intersections. 
Now the cone (190) touches the tangent plane to the conic node locus, viz. :— 
[«,/3]{[a, f]X + [a, y]Y + [*, £] Z} 
- [a, a] m f] X + [fi, y] Y + [(3, Q Z} = 0.(191), 
this being the form for the tangent plane to the conic node locus which can be 
deduced from (28), (29), and (76). 
Hence, to find the line of contact, whose equations are 
X/X' = Y/Y' = Z/Z', 
the origin of co-ordinates being taken at the singular point, 
f [f, f] X' + [(, Y' + [f, {] Z'J / {[a, /3] [a. f] - [a. a] [£, f]} 
= {[(, ,] X' + bl, V\ Y' + h, {] Z'} / {[a, /3] [a, ,] - [a, a] [fi, ,]} 
= {[?. a X' + [rj, {] Y' + [{, {] Z'J'/ {[a, ff] [a, Q - [a, a] [ft £]} . (192). 
