OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 551 
whence 
1 il 1 2) 9 
i +5 )=te +m), 
Vv Vy 
or 
— 4py, =v? + v,?. 
Hence we deduce that 
l= — 3m’. 
The elimination of € by help of the equation 
3 (E/v)* — 2k (Ev)? — 2n = 0, 
gives 
8192n (kK? — 2n)? — 384km* (18n — hk?) — 243m8 = 0. 
These two equations, connecting k, 1, m, n, define the second congruency of 
bitangents to the surface. 
A third is given by combining the equations 
y+ pv, +r? = 4m(v +n), 
vv = — Zl (v +»), 
Bt + Bef = Hs 
PE t/Pn? = — 4, 
EP + p= kEFL 
SW ale at ay)" = kg, + 1. 
The result of elimination may be expressed by saying that the expression 
256u? + 9u? (641m — m*) + 18u (21m? — Im*) — 97? (m3 — 31)? 
contains the expression 
27u? — 2u(k®? — 18kn) — 2n (k? — 2n)? 
as a factor. 
The surface has a nodal curve and a cuspidal curve. These are found by expressing 
the conditions that the equations 
y= 2) +P 
z= va + dat) 
solved for v, may have a pair of common roots. ’ 
If the roots are different we have a nodal curve, to wit, the curve traced by the 
point 
(405, £249, 4948) 
for different values of ft. 
