32 
ON A LOCUS DERIVED FROM TWO CONICS. 
[389 
1 (171 1)' 
and considering successive values of X; first the value X = — , = ————— > we have 
(m+ —) 
V mj 
_ A(m—l)fm—— 
(in -\')X 2 = ^ (rtl + 1) - m + 1 ± Y — 
m4— ml— 
m V m 
(m +1) ^ — 2^ + \ (m — 1) ^m — — 
7 I 
( in H— 
V w 
or observing that 
m 
(m + l)(m + -- 2) = (m+l)-(m-l) 2 = -(m-l)(m 2 - 1) = (in - 1) (m --), 
m 
in 
this is 
(m — 1) ( m — 
(m — X) x 2 = 0, or 
m 
1 
in H 
m 
or, what is the same thing, 
(m — 1) (m 3 + 2m 3 — 1) ^ or 
m m + 
¿) 
m/ 
1 
m H 
m 
x 2 = 0, or 
in- -„I in 
m 2 ) 
m 3 4- 2m 2 — 1 
The next critical value is X = m. The curve here is 
that is 
that is 
(x 2 + my 2 — 1) (mx 2 + y 2 — 1 ) — in (x 2 + if — cl) 2 — 0, 
m (x A + y*) + (1 + m 2 ) x 2 y 2 - (m + 1) (x 2 + i/) + l 
— in (x* + y 4 ) — 2m x 2 y 2 + 2ma (x 2 4- if) — ma 2 = 0, 
(m — l) 2 x 2 y 2 + (2ma — in — 1) (x 2 + y 2 ) + 1 — ma 2 = 0, 
or, substituting 
for a its value, 
2 ma — m 
2 in 2 + 2 
m + 1 
— (m + 1) = 
(m — l) 2 
^T+ 1 ‘ ’ 
the equation is 
, 1 (m 2 +l) 2 
(m — l) 2 (in 2 + m + 1) 
m (m +1) 2 
m (m +1) 2 
in 2 + 111 + I 
^f+ m+1 (^+y 2 ) 
- =0 
m (m + l) 2 
or, as this may also be written,