CURVES WHICH SATISFY GIVEN CONDITIONS. 
407] 
285 
*(2,1,1,1)= 
\m 3 n — 3m 2 n 2 — 3 mn 3 — 
4- 4£ra 2 /i + 2 4tmn 2 4- yn 3 
-Î* 
-ÎB 
-2D 
- D' 
=-im 4 
+ biv 
low 3 — 
%m-n 
— §m?n 
— \mPrP 
4- Jfbn 3 + y-m 2 n — ^-mn 2 
— §m 3 + im 2 B 4- 3 mn 2 
3 m 3 
— 37mn— 37 n 2 
4- fm 2 + fmn — Afn 2 
— ym 2 4 ymn + ^n 2 
+ 24m 2 — 3mn — -^n 2 
— 21m 2 
fn 3 
- -y-n 2 
— fm 4 — fm 3 7i — ym 2 n 2 — 3?nn 3 — f n 4 4- 4?n 3 4- ymîi 4- y-mn 2 4- 9n 3 — y-m 2 — 14?/w? — ^f-% 2 
(1) 
(2) 
(3) 
0) 
(5) 
(6) 
(7) 
(1) 
(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
4- 22571 4- öl to 3 + m 2 77 4- 7?m 2 + f?i 3 
— i m + f n + a ( 
4- ym — â §-n 4- « ( 
— i|pm 4- y n + a ( 
4- 36?n + a ( 
4- 1871 4-a ( 
— |m 2 — |m7î 4- 27i 2 4- y-m — — 75) + a 2 (— fm — f7i 4- y) 
m 1 
771“ 
m 2 
— m+ 8?i— l)4-a 2 ( 
4- |m?i — 1077i — 10?7 4- 40 ) 
— 9?n 4- 20) 
4- 7m - 12) 
— \n 2 4- \n — 6) 
- f) 
ym + 4 y7i 4- a {j(w? 4- m 2 n 4- mn 3 4- f7i 3 — f m 2 — mn + ^n 2 — fm— 4fn — 34)+ a 2 (— fm — f 77 + 13) 
(twelfth equation)’ 
129. We have consequently, by means of the results just obtained, 
Supp. (5) = *(5) 
+ N 
(first equation). 
Supp. (4, 1) = *(3,1) 
4“ 2 J 4- R 
(second equation). 
Supp. (3, 2) = *(3, 2) 
4- 6K + L + 3JV 4- 20 
(third equation). 
Supp. (3, 1,1) = *(3, 1, 1) 
+ D+E+F+ 2G + 3J + J' 
(fourth equation). 
Supp. (2, 3) = * (2, 3) 
+ Q 
(fifth equation). 
Supp. (2, 2, 1) = * (2, 2, 1) 
4- 36r +1 + 4/ 4- 2J' 
(sixth equation). 
Supp. (2, 1, 1, 1)= * (2, 1, 1, 1) 
+ Z? + 4(7 4- 4Z) 4- i)' 
(seventh equation).