752 
PROFESSOR CAYLEY’S ADDITION TO MR. ROWE’S MEMOIR. 
then this expression is 
y—tn r m r n s iJi s -\-^tn z mix—^%nm—\%n—\n +1, 
s>r 
or what is the same thing, for n writing its value S«/x, 
y= Sn,m r n s fx s + ^%n 2 vip — \tnm——\%n-\-l, 
s>r 
where in the first sum r, s have each of them the values 1, 2, . . . k, subject to the 
condition s > r ; in each of the other sums n, to, and /x are considered as having the 
suffix r, which has the values 1,2 , k. 
It is a leading result in Riemann’s theory of the Abelian integrals that y is the 
deficiency (Geschlecht) of the curve represented by the equation y) = 0 : and it 
must consequently be demonstrable d ‘posteriori that the foregoing expression for y is 
in fact = deficiency of curve y(x, y) = 0. I propose to verify this by means of the 
formulae given in my paper “ On the Higher Singularities of a Plane Curve,” Quart. 
Math. Jour., vol. vii., pp. (1866) 212-222. 
11 % 
It is necessary to distinguish between the values of — which are >, =, and < 1 ; 
and to fix the ideas I assume k=7, and 
m, TO, TOo 
—, —", — each > 1, 
AR AR AR 
—=1; say to 4 ,=/x 4 = \; and n±—Q, 
AR 
TO 5 
vp 
TO 
5 
AR 
At 
each <1, 
but it will be easily seen that the reasoning is quite general. I use S' to denote a 
sum in regard to the first set of suffixes 1, 2, 3, and %" to denote a sum in regard to 
the second set of suffixes 5, 6, 7. The foregoing value of n is thus 
n = %'np ~f Xd-f- 2"n/x. 
Introducing a third coordinate z for homogeneity, the equation y(x, y) — 0 of the 
curve will be 
where it is to be observed that ( ) ni '* 1 is written to denote the product of iq/q different 
»i «ii . . . 
series each of the form yz^~ l — Apc^ . . . ; these divide themselves into n x groups, each 
a product of series; and in each such product the /q coefficients are in general the 
