528 
ON THE HIGHER SINGULARITIES OF A PLANE CURVE. 
[374 
or writing 2 = 1, Y= 1, we find A = 1, and therefore 
X = pAxP~ l + qBx' l ~ x + ..., 
Z=—y—pAcc p — qBx q + ... ; 
here substituting for —y its value = Ax? + Bx q + ..., we have 
X = pAx p ~ x + qBafl~ x -f ... , 
Z = {l—p)Ax p + (1 — q) Bx q +.... 
Hence writing pAx p ~ x = 6, the equations are 
g-i 
x = e-B"d p ~ 1 -..., 
z= - A'd p ~ x - b'Op- 1 
so that eliminating 6, we have 
Z= 4T^ + £ , I^ ri + ..., 
and it is easy to see by Lagrange’s theorem, that the general form of the exponents 
in the series on the right-hand side is —B) + • • • ^ where f g,... are 
p l 
positive integers, zero included. The equation in line-coordinates being known, the 
subsequent investigation is precisely the same as that for the point-coordinates, and 
hence in the case of one branch, if this be in regard to its tangents /3-ic, and have ^iV 
common tangents with itself, then 2t' + Si=X, ¿' = /3 — 1, or, what is the same thing, 
t = % [N— 3(/3 —1)], ¿'=/3 — 1. The investigation in the case of a simple singularity of 
the values of S', k, t, l is thus completed.