PROFESSOR CAYLEY’S ADDITION TO MR. ROWE’S MEMOIR. 
75 7 
seqiiently the number with all of them is =——— [n 1 (m 1 — fa)— l]. But we obtain 
m i — AL 
this same number from each of the —/a x ) partial branches, and thus the whole 
071/ 
number is /x 2 ) -1 —[^(toi—/ iq) — l], =n 1 m 1 [n 1 (wi 1 —— l]. 
m i~fh 
Taking account of the other sets, each with itself, the whole number of such inter¬ 
sections is 
which is 
— /x 1 ) — 1 ] -(- n, 2 m. 2 [n. 2 (m. 2 —/x 3 ) —■ 1 ]+n 3 m 3 [w 3 (m,—/x 3 ) — 1 ], 
— %'n z m 2 — %'n 2 ni[x — %'nm. 
Observe now that — > — that is — < and that, these being each < 1, we thence 
fi x fx. 2 Wj m 3 
have 1 — ' w ‘> 1 — that is — 1 1 > : and we thus have 
/a, m 0 7771 77?o 
m. 
< 
nu 
< ——. 
071’-^ jJL i 071 o jJj.') 07l§ fl.^ 
m x \ n ) 
Considering now the intersections of partial branches of the two sets (z — 
( in, \ w,(ro a —|u.' a ) m , 1 
z —) respectively, a partial branch z—gives with each partial 
OTh 
branch of the other set a number =——1— ; and in this way taking each partial branch 
of each set, the number is n 1 (ra 1 — /x^.n^m^ — fx. 2 ). — 7 Hl _ } =7? 1 7n 1 n 2 (m 2 — fx 2 ) ; and thus 
for all the sets the number is 
= npn 1 %(mo — ,a 3 ) + —/x 3 ) + n. 2 m 2 n s (m 3 —/x 3 ), 
which is 
= Y n,m;n s m s — Y n r m r n a fi s , 
*>r 
where in the first sum the %' refers to each pair of values of the suffixes. But the 
intersections are to be taken twice ; the number thus is 
= 2 Hn r m r n s m s — 2Y n r m r n s [x s . 
s>r 
Adding the foregoing number 
^ / O 0 ^ / 0 v* * 
Y n~m~ — Y n-'m/x — Y nm, 
the whole number for the singularity in question is 
= (Sumy- — Y 'nm— Sn~m/x — 2Sn r m r n s jx s . 
s>r 
