407] 
CURVES WHICH SATISFY GIVEN CONDITIONS. 
283 
Q = ?? 2 + 8 m — ?? — 3a 
?? 2 + 8m — ?? — 3a (fifth equation). 
3G = 3mn 2 + 24m 2 — 3mn — 12n- — 96m + 12?? + a ( — 9m + 36) 
+ I — — 3mn 2 + 9mn -f 12?? 2 — 36?? + a (mn — 3m — 4n + 12) 
+ 4J = —12m 2 + 36??? +a( 4??? —12) 
4-2J' = — 6?? 2 + 18?? + a( 2n— 6) 
12m 2 + 6mn — 6?? 2 — 60??? — 6?? + a (mn — 8m — 2n + 30) 
(sixth equation). 
J5 = £m 3 ?? — 2m 8 — fm 2 ?? + 4m?? 2 + 10 m 2 — 14 m?? — 16?? 2 — 8m + 64 n (i) 
+ 4(7= m 2 ?? 2 4-8 m 3 — m 2 ?? —9????? 2 — 7 2 m 2 + 9????? + 20?? 2 + 160??? — 20?? (2) 
+ 477= —6m 3 + 42 m 2 — 72???, (3) 
+ U = — f ?? 3 + -^-?? 2 —18?? O) 
^??? 3 ?? + m 2 n~ — I??? 2 ?? — 
5 ????? 2 — f?? 3 — 20??? 2 — 5 mn + f?-n 2 + 80m + 26?? 
(i) 
+ a ( — 1????? 
+ 6???+ 6??—24) 
(2) 
+ a (— 3??? 2 
+ 27m — 60) 
(3) 
+ a ( 2??? 2 
— 14m + 24) 
(4) 
+ «( 
^-?? 2 —%n+ 6) 
(5) 
+ a (— m 2 — 1????? 4- £?? 2 4- 19??? 4- f?? — 54) (seventh equation). 
— = m 
+ |0- - 
4?? + 
m — 4?? + a (eighth equation). 
(2, 3) = — 4m — 4?? — 6 + 3a 
+ 2(4,1)= 2m+2??~12 
— 2??? — 2?? — 18 + 3a (used infra) 
— 2P 
- Q 
-2R 
+ J' 
= — 2??? 2 
+ 2m — 16?? 4 a ( 
6) 
= 
— n 2 — 8m + ?? + a ( 
3) 
= 
6????? — 18?? 4- a (— 2??? 
+ 6) 
= 
— 3?? 2 + 9?? 4- a ( 
??- 3) 
1(2, 3) + 2 (4, 1)) = 
6m?? 4- 6?? 2 + 54?? + a (— 2m 
-11?? -18) +3a 2 
— 2m- 4- 12m?? 4- 2n 2 — 6??? + 30?? + a (— 4??? — 10?? — 6) 4- 3a* 
(ninth equation). 
36—2