175 
578] A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
3 
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, 
V 
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, [Ges. Werke, 
t. I., pp. 77—79]) that we can in fact express x, y, that is, u s , v s , rationally in terms 
of the parameters a, /3 connected by the equation 
a 3 = 2/3 (1 + a + /3), 
which is that of a general cubic (deficiency = 1); in fact, we have 
that is, 
where a, /3 satisfy the relation just referred to. The actual verification of the equation 
IV. by means of these values would be a work of some labour. 
79. In the general case p an odd prime, then in I. we have at the origin one dp 
(in the nature of a fleflecnode) and at infinity two singular points each = ^(p — l)(p — 2) dps. 
I infer, from a result obtained by Professor Smith, that there are besides (p — l)(p — 3) 
dps; but I have not investigated the nature of these. And the Table of dps and 
deficiency then is 
I. l+(p-l)(p-2)+ (p— l)(p —3) 
II. 1 + (p — 1) (i> — 2) + ^ (p — 1) (p — 3) + £ (p 2 — 1) 
III. 1 + (p - 1) (p - 2) + \(p - l)(p- 3) + ±(p 2 - 1) + HP 2 -!) 
IV. l+(p —l)(p-2) + £(p —l)(p —3) + £(p 2 — l) + KP 2 —im(P 2 “l) 
dps. 
Def. 
2p 2 — 
7p 
+ 6, 
4p — 
5, 
2p 2 — 
5p 
+ 4, 
2p — 
3, 
2p 2 — 
4p 
+ 3, 
p- 
2, 
2p 2 — 
\P 
+ f, 
hp- 
f 5 
viz. his values of the deficiencies being as in the last column, the total number of 
dps must be as in the last but one column.