190 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
From (70) and (71) either 
x — a — y — b . 
or 
a (x — a) + a (y — b) — 2 m . 
(73) , 
(74) . 
(i.) Taking (73), and eliminating 2 and ( y — b) from it and (69), (70), it follows 
that 
(x — a ) 3 (x — ad- 4m/3a) = 0. 
Now, if x — a = 0 , then y — b = 0 by (73), and 2 = 0 by (69). 
Hence (72) is not satisfied. This solution corresponds to the biplanar node locus. 
But if 
x — ci— — Am/Za, 
y — b = — 4m/3a by (73), 
and 
2 = — 4 m /9 a by (70). 
These values satisfy (69)-(72). 
Hence 2 = — 4m/9a is an envelope. Each point £, 77 , — 4m/9a on it is the point of 
contact of one surface of the system whose parameters are 
a = f + Am/Sot, b = 7 ] + 4?n/3a. 
Hence 9 a 2 fi- 4m = 0 is an ordinary envelope. 
(ii.) Taking (74), and eliminating (x — a) and (y— b) from it and (69), (70), it 
follows that 
3 a 2 2 2 — 6 maz -f- 4m 2 = 0.(75). 
The corresponding values of (x — a), (y — b) are determined by (74) and 
a (x — a ) 2 — 2 m (x — a) — 2 mz -f- 4m 2 /a = 0 . 
These values satisfy (69)-(73). 
Hence each point rj, £ on the imaginary locus (75) is the point of contact of two 
surfaces of the system, whose parameters a, b are determined by the equations 
a 2 (£ — a ) 2 — 2ma (£ — a) — 2ma£ + 4m 2 = 0, 
a (£ — a) fi- a (y — b) — 2 m = 0 , 
where £ is one of the roots of (75), 
This accounts for the factor 
(3a 2 2 2 — 6 maz d - 4m 2 ) 2 
in the discriminant. 
