402] 
ON A SINGULARITY OF SURFACES. 
12 7 
Consider next the surface 
(1, h*, c, f g, h\y-z 2 , yz-x, xz - y 2 ) 2 = 0, 
being as already remarked, the general surface referred to a pinch-point as origin. 
Section by the non-special plane z = 0. 
The equation is 
(1, c,f, g, h$y, -x, — y 2 ) 2 = 0, 
where, attending only to the terms of the lowest order, we find (1, h, h 2 ][y, -x) 2 = 0, 
that is (y - hx) 2 = 0, showing that the origin is a cusp. 
Section by the non-special plane through the tangent line, viz. the plane y — 0. 
The equation is 
(1, h 2 , c, f, g, K$-z 2 , -x, xz) 2 = 0, 
or what is the same thing, 
that is 
h 2 x 2 -f 2hxz 2 — 2fx 2 z + cx 2 z 2 — 2gxz 3 + z 4, = 0, 
(hx + z 2 ) 2 — 2fx 2 z + cx 2 z 2 — 2gxz 3 = 0, 
2 f 2 q 
writing hx = — z 2 -f Ax*, we find at once fi = -1, and then A 2 = ~^ — -^-, so that the 
branches are hx = — z 2 + Ax*; whence we have at the origin a cusp of the second 
order or node cusp. 
Section by the osculating plane x = 0. 
The equation is 
(1, h 2 , c, /■, g, hjy-z 2 , yz, -y i ) 2 = 0; 
writing y = z 2 — hz 3 + Az*, we easily find = and then 
(1, h 2 , c, f, g, h][- hz 3 + Az 1 -, z 3 - hz 4 , - Af = 0, 
where the terms in z 6 , and disappear of themselves, the terms in z 1 give 
A 2 + 2gh = 0, and the branches are 
y = z 2 — hz 3 ± Az* &c., 
viz. there is a cusp of a superior order. 
Section by the tangent plane y = hx. 
The equation is 
(1, h 2 , c, /, g, li$hx — z 2 , -x + hzx, xz - h 2 x 2 ) 2 = 0,