128 
ON A SINGULARITY OF SURFACES. 
[402 
representing the projection on the plane of zx. Developing, the equation is 
2(f~9 h ) x- (h 2 x — z) + aAch 4, 
+ a?z — 2 h 2 (hf+ c) 
+ x 2 z 2 (h 4 + 2g 2 h 2 + 2fh + c) 
+ xz s — 2 (h 2 + 
+ ^ 1 =0, 
and there is at the origin a triple point (= cusp + 2 nodes) arising from the passage 
of an ordinary branch through a cusp; the tangent at the cusp being it will be 
noticed the line x = 0, that is the tangent x = 0, y = 0 to the nodal curve at the 
pinch-point. 
The results of the investigation may be presented in a tabular form as follows: 
Nature of Section. 
Plane of Section. 
Origin, an ordinary point. 
Origin, a Pinch-point. 
Non-special. 
Node. 
Cusp. 
Ditto, through tangent line 
of nodal curve. 
Tacnode = 2 nodes. 
Node-cusp, = node + cusp. 
Osculatingplane of nodal curve. 
y — z 2 + z 3 + &c. 
y =z i + hz* ± Az? Ac. 
Either of the two tangent 
planes. 
The single tangent plane. 
Triple point, one branch touch 
ing the tangent of nodal line. 
Triple point, = cusp + 2 nodes; 
the cuspidal branch touching 
the tangent of the nodal line. 
I have not considered the special cases where one of the two tangent planes, or (as 
the case may be) the single tangent plane of the surface coincides with the osculating 
plane of the nodal curve.