OF ELLIPTIC FUNCTIONS. 
455 
viz. writing u to denote an eighth root of — 1, (++1 = 0), then a dp is y=u\ To 
verify this observe that these values give 
6 + = + 6 
+ 20 +/ -20 
- 10 */ -10 
+ % +4 
-20 +/ +20 
- 6 / = + 6 
+ 10 x 4 y —10 
-20+/ -20 
+ 4* +4 
-20+/ +20 
or the derived functions each vanish. Thus I. has in all 1 + 12 + 8, —21 dps. 
In II. we have in like manner 1 + 12 + 4, =17 dps; viz. instead of the 8 dps, we have 
the 4 dps *=+, ^=+,(++1 = 0), or, what is the same thing, x=u, y=—u, where 
+ + 1=0. But we have besides the 12 dps given by 
x 3 —y 3 +5xy(x—y)=0, 1— +/=0, 
viz. we have in all 1 + 12 + 4 + 12, =29 dps. 
In III. we thence have 1+12 + 2 + 6, =21 dps; and, besides, the 12 dps given by 
+ + 1 5 x 2 y + 1 5 */ +/= 0 , 8 — 5+ + 10 , 27 / — 5 / + 8 +/ = 0 , 
in all 1+12 + 2 + 6 + 12, =33 dps. 
And in IV. we thence have 1 + 12+1 + 3 + 6, =23 dps; and, besides, the 12 dps 
given by 
++65 5 x 2 y +65 5xy 2 +/— 6 40*y — 6 40+/ = 0, 
— 256+32 0* +320^— 7 0+— 66 Oxy — 7 0/ + 320+;/ + 320#/ — 2 5 6+/ = 0 
(these curves intersect in 16 points, 4 of them at infinity, in pairs on the lines x=0, 
y— 0 respectively; and the intersections at infinity being excluded, there remain 
16 — 4, =12 intersections); there are thus in all 1+12 + 1 + 3 + 6 + 12, =35 dps. 
Or arranging the results in a tabular form and adding the values of the deficiency, 
we have 
dps. 
dps. 
Def. 
I. 
1 + 12 + 8 
= 21, 
= 15, 
II. 
1+12+4+12 
29, 
7, 
III. 
1+12+2+ 6+12 
33, 
3, 
IV. 
1+12 + 1+ 3+ 6 + 12 
35, 
1, 
so that the curve IV. is a curve of deficiency 1, or bicursal curve. It appears by 
Jacobi’s investigation for the quintic transformation (Fund. Nov. pp. 26-28) that we can 
in fact express x, y , that is u 8 , +, rationally in terms of the parameters a, connected by 
the equation 
+=20(l+«+0), 
which is that of a general cubic (deficiency =1) ; we in fact have 
2 — « v 4 u 5 
st — 2/3 w 4 ’ ^ v ’ 
