754 
PROFESSOR CAYLEY'S ADDITION TO MR. ROWE’S MEMOIR. 
n 5 (fi 5 —m 5 ) partial branches z—that is w 5 (/i 5 —m 5 )partial branches z = Au/^-mq- ... ; 
with in all w 5 (/x 5 — m 5 ) distinct values of A 5 : and the like as regards the unexpressed 
factors with the suffixes 6 and 7. 
Thirdly, singularities at the 6 points (z= 0, y — Ax=0), A having here 6 distinct 
values, at any one of which the tangent is y—Ax—h), and which are denoted by the 
function 
. but in the case ultimately considered \ is = 1 ; and these are then the 6 ordinary 
points at infinity, (z=0, y —Aa;=0). 
According to the theory explained in my paper above referred to, these several 
singularities are together equivalent to a certain number S'+A of nodes and cusps, 
viz., we have 
8' = Pt—fS(a-l) 
A= %-l), 
hence 
S'+A=iM-P(a-l) 
and (assuming that there are no other singularities) the deficiency 
l(K-l)(K-2)-SW 
is 
=i(K-l)(K~2)-iM+i%-i) 
this should be equal‘to the before-mentioned value of y, viz., we ought to have 
(K — 1) (K — 2) — M + 2 (a. — 1 ) = 2 £ n r m r n s y s -fi % nhi i y — %nm — 'tny — %n -fi2 
s>r 
or, as it will be convenient to write it, 
M — Iv~ — 3 lx -|-S(a — l) — '2%Tt r m/n s jx s — 
$>r 
which is the equation which ought to be satisfied by the values ol M and S(a l) 
calculated, according to the method of my paper, for the foregoing singularities of the 
curve. 
We have as before 
K = S'nm fi- %"ny + 0\. 
The term ^,?i r m r n s [i s , written at length, is 
