OF ELLIPTIC FUNCTIONS. 
451 
III. (if + Qxy -j- x 1 )" — 1 $xy(xy -j- 1 ) 2 = 0, that is 
y 4 +6afy*+a,’ 4 — ixy{^x 2 y 2 — 3# 2 — 3?/ 2 +4)=0 ; 
IV. (y~ + 6xy + off — 1 §xy(ixy — 3a’ — 3 t/ + 4) 2 = 0, that is 
y i — 7 6 2x 2 y 2 + x i — 4 xy 1 6 4 x 2 y 2 — 9 6x 2 y — 9 Qxy 2 + 33^ 2 -f 33^/ 2 — 96x— 96^+G4| =0, 
where it may be noticed that the process is not again repeatable so as to obtain a sextic 
equation between x, y standing for u 16 , v 16 respectively. 
The curve I. has a dp (fleflecnode) at the origin, viz. the branches are given by 
y 3 —2x=0, —x 3 —2y~0 ; and it has 2 cusps at infinity, on the axes x=0, y= 0 respec- 
tively; viz. the infinite branches are given by y-\- 2x 3 =ti, — #+2y 3 ==0 respectively. 
These same singularities present themselves in the other curves. 
The curve II. has the four dps (x 2 — y 2 =0, xy— 1 = 0), that is 
(x=y=l), (x=y=— 1), (x=i, y=—i ), (x=—i, y—i). 
Corresponding hereto we have in the curve III. the 2 dps x—y= 1, x=y= — 1, and in 
the curve IV. the dp x=y= 1. 
The cuiwe III. has besides the 4 dps y 2 -\-Qxy-\-x 2 —0, ^ + 1=0, that is 
(l+v/2, 1-^2), (1-./2, 1+^/2), (-1-^2, -l+v/2), (-l+x/2, -1-V2); 
and corresponding hereto in the curve IY. we have the 2 dps 
(3+2^72, 3-2^72), (3 — 2^/2, 3+2^72). 
The curve IV. has besides the 4 dps y 2 -\-§xy- b# 2 =0, ^xy— 3x— 3y+4=0, or say 
(2x— %)[2y— f)+|-=0, 2(^+f) 2 4-2(3/+f) 2 — i |- z =0. Hence the 4 curves have respec- 
tively the dps and deficiency following : — 
dps. dps. Def. 
2, 1 =3 7 
2, 1, 4 =7 3 
2, 1, 2, 4 =9 1 
2, 1,1, 2, 4 = 10 0; 
viz. the curve IY. representing the equation between u s and v 8 is a unicursal sextic. 
It may be noticed that except the fleflecnode at the origin, and the cusps at infinity, 
the dps in question are all acnodes (conjugate points). 
75. The foregoing equations may be exhibited in the square diagrams: — 
I. II. III. IV. 
y 1 y 3 y 3 y i t y 3 y* y 1 y* y 3 y 3 y i f y 3 y 3 y i 
=(y+i) 3 (y-J) =(y-i) 1 =(y-i) 4 =(y-i)‘ 
