OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 549 
If now we take the other value of x given by the quadratic x? = pv? we find for 
the other point of contact 
2 = — 3% (2¢ — e)), 
= — yyy (18¢ — 137”), 
au 9 
Z= Sage i0c <= 110°). 
The elimination of v from these equations gives the locus of the other point of con- 
tact. 
There is no second singular solution. 
The equation to the surface which has the lines for bitangents is to be found by 
eliminating p, v from 
It is 
31252 — 9000x7yz? + 8100x>yez + 2592x42° — 19440%y? — 2592x°y2z4 + 648y%23 = 0. 
Any point of this surface may be represented by the coordinates 
/ 
(a BoA the) 

The direction cosines of the tangent plane are proportional to 

Hence if the line 
y=ke+l 
is a tangent to the surface at this point, we have 
Eta she tl, 
v 

ve+h E = mé Fe N, 
ws 
pe — : = — ké + mv’, 

and therefore 

