524 
ON THE HIGHER SINGULARITIES OF A PLANE CURVE. 
[374 
axis of y not being a tangent to either branch, so that the exponents are none of 
them <1. If in the series for yi — y 2 , (the difference of the two ordinates) the least 
exponent is = P, then (whether P is integer or fractional) the two partial branches are 
said to have, at the given point, P common points, or, more briefly, to intersect in 
P points. We may from this definition calculate the number of intersections of two 
branches with each other, or of a branch with itself; for instance, suppose that at 
any point of the curve we have (a = 6) the sextic branch 
y — x^ + x* + ..., 
we have the six partial branches 
y x = x* + o$ + ... , y 4 = x* — x% + ..., 
4 5 4 5 
y 2 = ft) X s + X* + ... , y 5 — (O X s — x* + ... , 
y 3 = co 2 x* + x* + ..., y 6 = w 2 x$ — x* + ... ; 
hence calculating (what is most convenient) twice the number of intersections of the 
branch with itself, the partial branch y x intersects the other partial branches in 
f, •§, f, f, f points respectively, giving the sum ^- + 1 = ^; each other partial branch 
intersects the remaining five branches in the same number of points; and therefore 
twice the number of intersections is =47. 
For the singularity y = x*+x%-\-... in question, I say that if this be equivalent 
as above to 8' double points, k cusps, t double tangents and l inflexions, then that 
the number 47 just obtained is the value of 28' + 3k, and moreover, that the value 
of k is k = a — 1 = 5; that is, we have 28' + 3k = 47 and k = 5 ; or, what is the same 
thing, 8' = 16; k = 5. For the determination of the numbers r, i, it is to be observed 
that the foregoing theory of branches is a theory of the points of a branch, by means 
of point-coordinates: there is a precisely similar theory of the tangents of a branch 
by means of line-coordinates, and we may inquire as to the number of the common 
tangents of two partial branches; and thence as to the number of common tangents of 
two branches, or of a branch with itself—it will appear that the line-equation of the 
branch is Z = X 4 ... 4- X - ... , so that the branch (which is as to its points sextic, 
a = 6) is as to its tangents quadric, ¡3=2, the two partial branches have with each 
other the number =^ 5 - of common tangents, or twice this number is =15; that is, we 
have 2t' + %i =15, and moreover ¿' = /3-1=1, that is t' = 6, ¿'=1; or finally for the 
singularity in question, the numbers 8', r', t are =16, 5, 6, 1 respectively. 
And so generally in the case of a branch which is as to its points a-ic, having 
with itself a number =^M of common points; and as to its tangents /3-ic, having 
with itself a number = of common tangents, we have 28’ + 3«' = M, k=ol — 1, 
2t' -f St = A 7 , ¿' = /3 — 1, or, what is the same thing, the values of 8', k', t, l are 
8' = \[M- 3(a -1)], 
k — a — 1 , 
T'=HAT-3(/3-i)], 
I = /3-1 .