456 
ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
that is. 
«*(=*)=0’(^), v\=y)= { 
where a, 0 satisfy the relation just referred to. The actual verification of the equa- 
tion IV. by means of these values would be a work of some labour. 
T9. In the general case p an odd prime, then in I. we have at the origin a dp (in the 
(p l)(w 2) 
nature of a fleflecnode) and at infinity 2 singular points each — ^ ; dps. I infer, 
from a result obtained by Professor Smith, that there are besides (p — \){p— 3) dps ; 
but I have not investigated the nature of these. And the Table of dps and deficiency 
then is 
I. 1+(p-1Xp-2)+ O— 1)Cp— 3) 
ii. i+{p-i)(p—2)+\{p-i){p-'Z)+i(p*—i) 
III. 1+(P- l)(p-2)+i(p— l)(p-3)+i(p 2 — 1)+1(P 2 -I) 
dps. Def. 
2p 2 — 7p+6, ip — 5 
2p 2 -5p+i, 2p—S 
2p 2 —4p-{-3, p—2 
2p 2 -{p+l ip— I 
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. 
