758 
PROFESSOR CAYLEY’S ADDITION TO MR. ROWE’S MEMOIR, 
Similarly for the singularity (z — 
number of intersections is 
Ms \ it s(f l t -,ll s) 
; taking each set with itself, the 
5^o[ n 5 0*5- m s)- !] + — m e)-!] + ffi/rtW/k“ to t) ~ 1 1 
which is 
= 2 , n~j u,- 
n-vi/jL — z rifx. 
We have here — 5 >-- and each of these being less than 1, we have 1- 2 <1 — ", 
Mg Mg ' Ma Mg 
m- 
that is or Ms 
Mg 
Mg 
/X - 972/- 
; and so 
__Mt_ < Mg < % 
Mv —m 7 Me -m 6 Ms - ' m s’ 
/ Ms \it s (/x--ire 5 ) / __Ms_\'M s(Ms-''M«> 
Hence considering the two sets (z— y^s-m 5 j and (z—, a partial 
branch of the first set gives with a partial branch of the second set - G intersec- 
tions : and the number thus obtained is M 5 (p, 5 —m 5 ).?t 6 (p, 6 —m 6 ).-—, = n 5 n 6 /r 6 (/a- — m 5 ). 
Me -m c 
For all the sets the number is 
^6^6(M5-^5) + ^Vk/*7(^5- m 5)+^G ? kM7(/*6- TO 6) 
or taking this twice, the number is 
= 2 %"n r [x r n s jx a — 2 %"n r m r n s n s 
s>r 
where in the first sum the S" refers to each pair of suffixes. Adding the foregoing- 
value 
v^// o O >rA // O -W/ 
2 rr/r'—2 n^nifi — 2 n/x, 
the whole number for the singularity in question is 
s>r 
and the proof is thus completed. 
Referring to the foot-note ante (p. 753), I remark that the theorem deficiency, is 
absolute, and applies to a curve with any singularities whatever : in a curve which has 
singularities not taken account of in Abel’s theory, the quelques cas particuliers que 
je me dispense de considerer,” the singularities not taken account of give rise to a 
diminution in the deficiency of the curve, and also to an equal diminution of the value 
of y as determined by Abel’s formula ; and the actual deficiency will be = Abel’s y — 
such diminution, that is, it will be = true value of y. 
