507] 
CERTAIN QUARTIC CURVES BY A MODIFIED OVAL CHUCK. 
155 
giving four intersections. But the intersections by the line by + cz = 0 (that is, by any 
line through the point y = 0, z = 0) are obtained from the equation 6f £ + c£ 2 = 0 ; viz. 
this breaks up into 6£+c£=0, £=0, and the last factor combined with the equation 
of the circle gives £ = 0, f 2 + rf = 0, the two circular points at infinity, corresponding 
each to the point y — 0, z = 0: the other factor gives points corresponding to two 
variable points on the curve; that is, a line through the point y = 0, z — 0 meets the 
curve in this point twice and in two other points. Again, making 6 = 0, or taking the 
line to be the line at infinity z = 0, the equations then are £ 2 = 0, p + r/ 2 =0; viz. we 
then have the circular points at infinity each twice, corresponding to the point y = 0, 
z = 0 four times, and no other point; that is, the line z=0 meets the curve in the 
point y — 0, z = 0 four times. We thus see that the curve has at y — 0, z = 0, that is 
at infinity on the line y — 0, a tacnode (counting as two nodes), the tangent at this 
point being the line at infinity z = 0. The curve being trinodal has of course one 
other node.