446]
OF A NODAL BICIRCULAR QUARTIC.
185
which, in fact, show that the locus is a bicircular quartic. To put in evidence the
third node, I assume that the values belonging thereto are m = Mj, u — u 2 , and that
the coordinates of the node are a, ft ; we have thus
M, 2 — 1 , to 2 — m 2 2tom x n£ — 1 7 u£ — to 2 2tom 2
oc = - a „ — c —- ;—-, = — a „ . v + ft-r-: C
m x 2 + 1 u£ + to 2 lij 2 4- m 2
m 2 2 +1 Mo 2 + to 2 m 2 2 + to 2 ’
2 Mi 2mtq m x 2 — to 2 _ 2m 2 , 2mM 2 m 2 2 4- m 2
^ a Mj 2 + 1 + m 2 2 + m 2 + ° M 2 2 + to 2 ’ a u.£ + 1 m 2 2 + to 2 ° m 2 2 4- m 2 '
These give 6, c, a, ¡3 in terms of a, to, u 1} u 2 ; and we may then express the values
of x — a, y — ft in terms of a, to, u lt m 2 , u. I find
b =i f 1 + («,^+^¿+'1) [(!<1 + “ a)!+(1 “ ra)(1 ~ “- )] | ■
c =
to + 1
1 (V+WTT) [ -(“. + “=)(» -«,«,) ]p
and then
x = — a
u 2 — 1 a
to 4-1
+1 + m I" 1 + (V + 1)W + 1) [( “‘ + “ !)i + (1 - (1 - ^“ a)] f
№ f TO 4-1 r / w 2tom
+ m| («,’ + !) („/+!)[ + -«.»>) 1}^^.
2tt af-,
2/= + — 1— 1 +
TO + 1 r/ \o /i X /1 ,-d 2tom
[(tq + m 2 )- + (1 — to) (1 - Mj m 2 )]
m 2 + 1 to ( (m 2 2 4-1) (m 2 2 4- 1)
I®! m 4' 1 r / .\/ \ -|) m
+ ml («,« + l)(^ + l) [ -(». +«.)(“-»A) ]}-
a to 4-1 w x
a = — , „ , 1W 9 , -.x (1 — MiM 2 ) (to 4- ih m 2 )
to (u£ + 1) (u£ + 1) ' v y
m 2 4- to 2
m 2 — TO 2
+ to 2 ’
_ a to 4-1 / w .
^ = m (u,* +1) («,»+ 1)+ “O (« + «.%);
and then
(*-«) = - w+1)w + 1) ( tf+1)(M , + m>) X P - («- + rn) + (1 - rn) («, + ) »],
iy —ft) —— ( Wi 2 + 1 )( M2 2 + !) ( tt ». + i )(u* + m?) * ^ + u2 ) ^ U ~ + ^ 171 } ^ ~ UlU J u
where, of course, the factors (u — г¿ 1 ), (u — m 2 ) indicate the node (a, ft). We have moreover
(x rrl 2 i (v RY - 4 (to + l) 2 a? (m-m 1 ) 2 (m-m 2 ) 2
^ ^ (zq 2 + 1) (u.£ +1) ( a 2 4 1) (m 2 + to 2 ) ’
so that, writing
__ _ x-rx 1 [(1 — M]M 2 ) (m 2 + to) + (1 - m) (m x 4- u. 2 ) m]
(x — a) 2 + (y — ft) 2 2(to+1)m (m — Mj) (m - u£)
y — ft _ 1 [(mj + m 2 )(m 2 4- to) - (1 - to) (1 - M t M 2 ) m]
(m — u£) (u — m 2 ) ’
(a? — a) 2 4 (y — /3) 2 2 (to 4 1) a
C. VII.
24
