174 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578 
t 
where the subscript line, showing in each case what the equation becomes on writing 
therein x = 1, serves as a verification of the numerical values. 
78. The curve I. has at the origin a dp in the nature of a fleflecnode, viz. 
the two branches are given by a~ s +4y = 0, — y 5 + 4a? = 0 respectively; and there are 
two singular points at infinity on the two axes respectively, viz. the infinite branches 
are given by — y —4a? 5 = 0, x — 4y 5 = 0 respectively. Writing the first of these in the 
form — yz 4 — 4a 5 = 0, we see that the point at infinity on the axis x = 0 (i.e. the point 
z=0, x = 0) is =6 dps; and similarly writing for the other branch xz i — 4y 5 = 0, the 
point at infinity on the axis y = 0 (i.e. the point z = 0, y = 0) is =6 dps *. 
Moreover, as remarked to me by Professor H. J. S. Smith, the curve has 8 other 
dps; viz. writing gw to denote an eighth root of —1, (gw 8 + 1=0), then a dp is x = co, 
y = eu 5 . To verify this, observe that these values give 
CO 5 
CO 
Qx 5 = + 6 
— Qy 5 = + 6 
+ 20 x?y 3 — 20 
+ 10 x*y — 10 
— 10 xy* — 10 
— 20 x 3 y 3 — 20 
+ 4 y +4 
+ 4x +4 
— 20 x?y 5 + 20 
— 20 x 5 y 4 + 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 x= co 2 , y = go 2 , (go 8 + 1 = 0), or, what is the same thing, x = co, 
y — —co, where go 4 + 1 = 0. But we have besides the 12 dps given by 
x 3 — y 3 + 5xy {x — y) = 0, 1 — x 3 y 3 = 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 
a? 3 + 15x 3 y + 15xy 2 + y 3 = 0, 8 — 5a? 2 + 10xy — 5y 3 + 8x 3 y 3 = 0, 
in all 1+12 + 2 + 6 + 12, = 33 dps. 
And in IY. we thence have 1 + 12 + 1 + 3 + 6, =23 dps; and, besides, the 12 dps 
given by 
a? 3 + 655x 3 y + 655xy 3 + y 3 — 640xy — 640x 3 y 3 = 0, 
- 256 + 320a? + 320y - 70a? 2 - 660xy - 10y 3 + 320^y + 320xy 3 - №§x 3 y 3 = 0, 
(these curves intersect in 16 points, 4 of them at infinity, in pairs on the lines 
a? = 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. 
* These results follow from the general formulte in the paper “ On the Higher Singularities of Plane 
Curves, ’ Camb. and Dubl. Math. Journ. t. vn. (1866), pp. 212—223, [374]; but they are at once seen to 
be true from the consideration that the curve yz*- x 5 = 0, which has only the singularity in question, is 
unicursal; the singularity is thus = 6 dps.