374] 
ON THE HIGHER SINGULARITIES OF A PLANE CURVE. 
525 
I say that a singularity is simple when we have one branch, compound when we 
have more than one branch ; the case above considered is that of a simple singularity, 
viz. we have on the curve one point, one tangent, one branch. 
We may have a compound singularity where the branches all touch, that is we 
may have one point, one tangent, several branches. It may be seen that if \M denote 
the number of common points of all the branches (that is of each branch with itself, 
and of every two branches with each other), and in like manner if denote the 
number of common tangents of all the branches (that is of each branch with itself, 
and of every two branches with each other), then the formulae are 
S' = $[M-3S(a- 1)], 
k= 2 (a — 1) , 
r' = i[iY- 3208-1)], 
*'= SG8-1), 
where the signs 2 refer to the different branches. 
Again, we may have a compound singularity, one point, several tangents with to 
each of them a branch or branches; here if if denote the number of the common 
points of all the branches, and N the number of the common tangents of all the 
branches belonging to any one tangent, then the formulae are 
S'=i [if — 32'2 (a — 1)], 
«'= 2'2(a-l), 
T' = 4S'[i\r-32 08-1)1 
t '= 2-2 OS-1), 
where the signs 2 refer to all the branches belonging to the same tangent, and the 
signs 2' to the different tangents. It is to be remarked, that the point on the curve 
is equivalent to the 3' double points and k cusps; each tangent is equivalent to 
| [N — 32 (/3 — 1)] double tangents, and 2 (/3 - 1) inflexions, the numbers N, /3 referring 
of course to the tangent in question. 
Lastly, we may have a compound singularity, one tangent, several points (of 
contact), with to each of them a branch or branches; here if N denote the number 
of the common tangents of all the branches, if the number of the common points of 
all the branches belonging to any one point of contact, the formulae are 
S' = £2' [(if — 32 (a-1)1 
*'= 2'2 (ar — 1) , 
t' = 4 [(JV - 322' (/3-1)], 
»'= 2'2 (/3 — 1), 
where the signs 2 refer to all the branches belonging to the same point of contact, 
and the signs 2' to the different points of contact; it is to be remarked that the