194 
PROFESSOR M. J. M. HILL OH THE LOCUS OF SINGULAR POINTS 
i.e., 
^ ^ ^comes [f, f] + [f, a] | + [f, / 3 ] || 
[||] {[£, 0] + [«, /3] | + [/ 3 , / 3 ] || by (48), 
therefore S' 2 / (x, y, z, a x , bfjdx 2 vanishes by (80). 
In like manner S f (x , y, z, a. 2 , b^/dx 2 also vanishes when x = y = y, z — £. 
Now writing A = It//, for brevity, and forming all the differential coefficients up 
to the fifth order inclusive, each term in any of these differential coefficients is the 
product of terms, one of which is / or f 2 or a first or second differential coefficient 
of/i or/ 2 . 
Hence when x = y — y, z — £, all the differential coefficients up to the fifth order 
vanish, and, therefore, by Art. 1, Preliminary Theorem B, A contains U 6 as a factor. 
Example 6.—Locus of Uniplanar Nodes. 
Let the surfaces be 
a (x — a) 3 + /3 (y — bf J -f 3 [c (x — a) -f- t/z]'- = 0 
where a, /3, c, p' are fixed constants; a, b the arbitrary parameters. 
(A.) The Discriminant. 
The discriminant is the same as that of the equation 
aX 3 + /3Y 3 + 3Z (cX + gzZf = 0. 
Hence 
S = 0 
T = a/3 2 /z 3 (Qap'Z — 4c 3 ) 
A = a 2 /3t/z 6 (9agz — 4c 3 ) 2 . 
(B.) The Locus of Unodes is z — 0. 
For putting r = a -|- X, y =6 + Y, z = Z, the equation becomes 
aX 3 + /3Y 3 + 3 (cX + r/Z) 2 = 0. 
Hence the new origin is a unode. There are no other singular points on the 
surface. 
Hence the locus of unodes is z = 0. 
(C.) The Envelope such that every 'point on it is the point of contact of two non- 
consecutive Surfaces is 9agz — 4c 3 = 0. 
To prove this it is necessary to find the tangent planes parallel to z — 0. 
