32 
ADDITION TO MR ROWE S 
[713 
where observe that the expressed factor indicates n 5 branches iz — y^~ m A , or 
Ms 
say n 5 (/jL 5 — m 5 ) partial branches z — y^~ ms , that is, n 5 (/x 5 — m s ) partial branches 
Ms _ 
2: = 4-... 5 with in all n 5 (y, 5 — m 5 ) distinct values of ri 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 = 0, and which are denoted by 
the function 
but in the case ultimately considered X is = 1 ; and these are then the 6 ordinary 
points at infinity, (z = 0, y — Ax = 0). 
According to the theory explained in my paper above referred to, these several 
singularities are together equivalent to a certain number 8' + k of nodes and cusps; 
viz. we have 
8' = IM- |2(a-l), 
#' = 2 (a — 1), 
hence 
8' + k' = pi-|2(a-l). 
Assuming that there are no other singularities, the deficiency 
\{K-\){K-2)-8'-k! 
This should be equal to the before-mentioned value of 7; viz. we ought to have 
(K - 1) (AT — 2) — If + 2 (a — 1) = 2 %n r m r n g /j, s + Xn-m/x — 2nm — Xn/x — 2 n + 2, 
s>r 
or, as it will be convenient to write it, 
M=K- — 3K + 2 (a — 1) — 22n r m r n s fi s — Xn-m/x -f 2nm + 2njx + 2n, 
8>V 
which is the equation which ought to be satisfied by the values of M and 2 (a— 1) 
calculated, according to the method of my paper, for the foregoing singularities of 
the curve. 
We have as before 
K = X'nm -f 2"n/j, + dX. 
The term 2n r m r n s fx s , written at length, is 
s>r 
«i»«! (rc 2 /a 2 + 
n s fl :t + dX + H 5 fl 5 + 7) 6 fX 6 -f n 7 /x T ) 
-f n 3 m 2 ( 
n 3 /i 3 + dX 4- n 5 /x 5 + n B /x s -f n 7 /x 7 ) 
+ n s nio ( 
dX + n 5 /x 6 + n B fx B + n 7 fx 7 ) 
+ dx ( 
n 5 /X 5 + U B fX 6 + n T fX 7 ) 
+ n 5 m 5 ( 
n B /x B + n 7 y 7 ) 
+ n 6 m 6 (