756 
PROFESSOR CAYLEY’S ADDITION TO MR. ROWE’S MEMOIR. 
The complete value of %(<x — 1) is thus 
= %'nm—% "nm—%'np -f- %"n y—%'n — %"n — 6k—0. 
Substituting all these values we have 
M = (%'nm -f- %"nff -f- 2 6\(%'nm -\-%"ny) J r (6\) 2 
-3(%'nm-\-%"ny) — 36\ 
+ %'nm—%"nm—%'ny +2'bip —%'n — %"n -\-6X—6 
—2%n,m r n s y s —26\(%'nm-\-%"nix)—2%'nm.%"ny—2%''n r m r n s ii s 
s>)' s>r 
— %'rrm y — ( OX) 2 — % "n 2 my 
-f %'nn i + 6\ -f -%"nm 
■f%'ny-{-6\ J r%"ny 
-\-%'n-\- 6-\-%"n, 
or reducing 
M = (%'nm)' 2 — %'nm — %'rrmy —2%'n r m r n s y s 
s>r 
+ ( %"ny) 2 — %"ny—%"n 2 my — 2 %"n r m r ii s y s ; 
s>r 
and it is to be shown that the two lines of this expression are in fact the values of M 
belonging to the singularities (z— x m '~^) . . . , and [z—y^-^J . . . 
respectively. We assume X= 1, and there is thus no singularity ( y — x K ) . 
I recall that, considering the several partial branches which meet at a singular 
point, M denotes the sum of the number of the intersections of each partial branch 
by every other partial branch (so that for each pair of partial branches the inter¬ 
sections are to be counted twice). Supposing that the tangent is £=0, and that for any 
two branches we have z 1 = A 1 a^’ 1 , z 2 =A pf- (where p x , p., are each equal to or greater 
than 1), then if p 2 =p h and z L —— — A. 2 )x Pi where A]—An not=0 (an assumption 
which has been already made as regards the cases about to be considered), then the 
number of intersections is taken to be —p x ; and if p x and p. 2 are unequal, then 
taking p 2 to be the greater of them, the leading term of z x —z 3 is = A 1 x Pl , and the 
number of intersections is taken to be —p x ; viz., in the case of unequal exponents, 
it is equal to the smaller exponent. 
/ to , 
Consider now the singularity (z—cc m i-Mi) . . . ; and first the intersections of a 
Wi 
partial branch by each of the remaining —/x } ) — 1 partial branches of 
nv 
the same set: the number of intersections with any one of these is =- - —; and con- 
