PROFESSOR CAYLEY’S SUPPLEMENTARY MEMOIR ON CAUSTICS. 
15 
For m —< f >\/2, the loop has altered its form in such wise as to exhibit the node 
and two cusps, the curve has therefore two real nodes. 
For m=\/2, the two nodes unite together into a tacnode, so that the loop is on the 
point of disappearing ; and for m < \/2 > 1 the nodes are imaginary, and there is thus 
no longer any loop. 
In all the above forms the double tangent x=m touches the curve at the points 
y ~ ili hut the other two intersections of the double tangent with the curve are ima- 
ginary. 
