590 PROFESSOR A. C. DIXON ON THE SINGULAR SOLUTIONS 
If the straight line is a bitangent, these equations are also satisfied when other 
values &, v,; ave put in the place of €, v respectively. 
Hence v and », are either the same root, or different roots of the equation 
that is, 
ie 
or else 
VV) 3 
yptp,—- 7) 
at a p+op ee 
and 
VV" 3] 
——+ = 3], 
V+ Vy 
Also €/y and €,/v, are either the same or are different roots of the equation 
3 (E/v)* — 2k (E/v)? — 2n = 0, 
in which case 
g/v cae a 1/1, 
or else 
EB) + Et)n? =H, 
BE 2p? = — 3n. 
The solution 
ae (Ss & 
is irrelevant; the solution 
gives the original congruency, and 
pan, +E = ty 
gives 
A new series of bitangents is given by taking 
Ev = — &/y, v=. 
We have 
y+ vp, +»? = 2m (v +), 
yyy, = — 7l(v+y), 

