252 
PROFESSOR M. J. M. HILL ON THE LOCHS OF SINGULAR POINTS 
i.e., where 
a = x 1 + 
2 xy + mz 
)» b = yU - 
2 xy + mz 
Hence the points of contact are determined by 
Hence 
2 xy + mz — m — 0, 
a — x 
J 
( z 2 — to 2 ) (z — 1 ) — 2 dim = 0 . 
Hence when a, b are given, there are three values of z, and three corresponding 
values of x, and three corresponding values of y. Hence each surface touches the 
envelope at three points. But each point on the envelope is the point of contact 
of only one surface of the system, since when the coordinates x, y, z of the point of 
contact are given, the values of a, b, the parameters of the surface touching the 
envelope there, are determined by the simple equations 
a = x (1 + z/m), b = y (I — z/m). 
The result may be verified thus :— 
The values of x, y, z satisfying the equations 
a = x (I + z ! m ), b — y (1 — zjm), ( z 2 — m 2 ) («—!)— 2 abm — 0 . . (a), 
will satisfy at the same time 
and 
( bx — ay + z)~ — 2 3 — 2 mz (x — a) (y — b) = 0 
2 xy + mz — m = 0 
2 ( bx — ay + z) b — 2 mz {y — b) — 2 (bx — ay + s) a — 2 mz(x — a) 
2y 2x 
2 (bx — ay + z) — 3 z 2 — 2m (x — a) (y — b) 
m 
( 0 ). 
(r)- 
If #, y, z satisfy (a), then 
, , 2 (z 2 4- 2a&m — m 3 ) 
— ay + 2 = -s-o- 
■j z l — m“ 
