Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., late sadlerian professor of pure mathematics in the University of Cambridge (Vol. 9)

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
	        
Waiting...

Note to user

Dear user,

In response to current developments in the web technology used by the Goobi viewer, the software no longer supports your browser.

Please use one of the following browsers to display this page correctly.

Thank you.