452 
PEOFESSOE CAYLEY ON THE TEAN SFOEMATION 
where the subscript line, showing in each case what the equation becomes on writing 
therein x=l, serves as a verification of the numerical values. 
The curve IV. being unicursal, the coordinates may be expressed rationally in terms 
of a parameter ; and we in fact have 
«^ 2 +*> *(2 + a )3 
l + 2a ’ J (1 + 2a) 3 " 
These values give 
16 xy =16 a 4 (2 + «) 4 =(l + 2a) 4 , 
4:-\-4:xy—3x—3y= (4, 8, 12, 32, 50, 32, 12, 8, 4J1, a) 8 -=-(1+2 a) 4 , 
x 2 + Gxy+y 2 = 4« 2 (2 +a) 2 (4, 8, 12, 32, 50, 32, 12, 8, ijl, a) 8 -=-(1+2 a) 6 , 
and the equation of the curve is thus verified. 
76. Considering in like manner the modular equation for the quintic transformation, 
we derive the four forms as follows : — 
I. x 6 y 6 + 5 x % y%x 2 —y 2 ) + kxy(l — x 4 y 4 ) = 0 . 
II. \x 3 —y 3 +5xy(x—y)\ 2 —16xy(l—x 2 y‘ i y i =0, that is 
x 6 + 15 x 4 y 2 + 1 5 x 2 y 4 ~\~y 6 ~2xy(8— 5x 4 + 1 0 x 2 y 2 — 5 y 4 + 8 x 4 y 4 ) = 0 . 
III. (x 3 -{-15x 2 y-\-15xy 2 -\-y 3 ) 2 —4:xy(8 — 5x 2 -\-10xy~5y 2 -\-8x 2 y 2 ) 2 =0, that is 
# 6 + 6 5 5x 4 y 2 + 6 5 5#y +y 6 - 64 0 x 2 y 2 - 640 x 4 y 4 
+xy(- 256+320^ 2 + S20y 2 -70x 4 - 660^/- 70/+320^y + 320^y- 256^y) 
IV. (or 3 + 6 5 hx 2 y + 6 5 hxy 2 -\-y 3 — 6 4 0 xy — 6 4 0 x 2 y 2 ) 2 
-^(-256+320a’+320y-70^-660^-70^+320a’ 2 ?/+320^ 2 -256^y) 2 =0: 
or expanding the two terms separately, this is 
