36 ADDITION TO MR ROWES MEMOIR ON ABEl/s THEOREM. [713 
We have here — > —; each 0 f these being: less than 1, we have 1—— <1—— , 
/¿5 /¿6 /^5 № 6 
that is, or —>—and so 
fi 5 /¿6 fi 5 iu 5 fig — m 6 
/¿7 /As ^ /Ai 
¡1 7 — m 7 Ia 6 ~ w 6 yu-5 — m 5 ' 
Hence considering the two sets 
/ \ / M-fi \ w e (fx 6 - r«jj) 
(2 — y^~ m aJ and ( 0 — y^ e ~ me J > 
a partial branch of the first set gives with a partial 
intersections: and the number thus obtained is 
branch of the second set 
fh 
/A; Vie, 
n 5 (jjl 5 - m 5 ). n 6 (/x 6 - m 6 ). —, = n 5 n 6 fx 6 (fi B - m B ). 
/¿6 m 6 
For all the sets the number is 
n 5 n 6 fx 6 Os - m s ) + n 5 n 7 /x 7 - m B ) + n B n 7 fi 7 (fx 6 - m 6 ) 
or taking this twice, the number is 
= 2 %"n r fi r n 8 fi s — 2%"n r m r n s ix s 
s>r 
where in the first sum the 2" refers to each pair of suffixes. Adding the foregoing 
value 
2 "n 2 fi 2 — 2' 'rfm/jL — 2'V, 
the whole number for the singularity in question is 
= (2 /, Wya) 2 — 2"iifi — S"ri 2r mfjL — Tt"n r m r n s yug; 
s>r 
and the proof is thus completed. 
Referring to the foot-note (ante, p. 31), I remark that the theorem y = 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 particulars 
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.