172
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [_578
76. Considering in like manner the modular equation for the quintic trans
formation, we derive the four forms as follows:—
I. x e y 6 + 5x 2 y 2 (x 2 — y 2 ) + 4>xy (1 — x 4 y 4 ) = 0 ;
II. {x 3 — y 3 + 5xy (x— y)} 2 — 16xy (l — x 2 y 2 ) 2 = 0, that is,
x 6 + 15afy 2 + lox 2 y 4 + y 6 — 2xy (8 — 5a-' 4 + 10x 2 y 2 — by 4, + 8afy 4 ) = 0;
III. (a? + 15x 2 y + 15xy 2 -\-y 3 ) 2 — 4>xy(8 — 5x 2 + 10xy — oy 2 + 8x 2 y 2 ) 2 = 0, that is,
x 6 + 655ofiy 2 -f- 655x 2 y 4 + y 6 — 640 x 2 y 2 — 640a^y 4
+ xy (— 256 + 320a: 2 + 320y 2 — 70a^ — 660x?y 2 — 70y 4 + 320x 4 y 2 + 320x 2 y 4 — 256x 4 y 4 ) = 0 ;
IY. (a: 3 + 655x 2 y + 655xy 2 + y s — 640xy — 64*0x 2 y 2 ) 2
— xy (— 256 + 320a; + 320y — 70a? — 660xy — 70y 2 + 320a ,2 y + 320xy 2 — 256x?y 2 ) 2 = 0 :
or, expanding the two terms in the last equation separately, this is
xy
a?y
xy 2
a?y
x?y 2
X y 3
x 4 y
x?y 2
x*y 2
X y 4
X 6
x>y
x 4 y 2
a?y 3
x 3 y 4
X y 5
y 6
afy 2
X 4 }/
x 3 y 4
x 2 y 5
a 5 ?/ 3
x 4 y 4
a 3 ?/ 5
x’y 4
x 4 y 5
ofiy 5
- 65536
+ 163840
+ 163840
- 138240
+ 409600
- 542720
- 138240
1280
+ 44800
- 838400
+ 631040
- 838400
+ 631040
1280
+ 44800
+ 1
+ 1310
- 4900
+ 430335
- 297200
+ 1677252
- 986072
+ 430335
- 297200
+ 1310
- 4900
1
1280
+ 44800
- 838400
+ 631040
- 838400
+ 631040
1280
+ 44800
- 138240
+ 409600
- 542720
- 138240
+ 163840
+ 163840
- 65536