IV. 
But 
in 
On 
85), 
als) 
and 
îrott, 
that 
in 
the 
ntic 
c of 
= 0. 
to 
an 
actic 
ation 
ation 
for the equation y — f (x) = 0 of the curve is a = 0, that is = 0. Passing first to the 
form U, =(%, y, l) m = 0, we have 
dU^dUdy 
and thence 
viz. substituting for ^ its value from the first equation the condition = 0, becomes 
(d_U\* d?u_ 2 du du æu_ /duy d?Tj_ 0 
\dyj doc 1 dy dx dxdy \dx) dy 2 ’ 
and we can then make the further transformation to the form U = (x, y, z) m , = 0, and 
so obtain but not very easily the result H (TJ)= 0: but in the transformations for the 
sextactic point, besides the differential coefficient a of the second order we have the 
coefficients b, c, d of the orders 3, 4 and 5 respectively ; and the complication is thus 
very much greater. 
343, 354, 374. The principal paper is 374; 354 is a mere résumé of this; and 
343 relates to the higher singularity which first presented itself, and which is there 
shown to arise from the coalescence of a node and a cusp, but in 374 (where it is 
considered more fully) it is shown to be equivalent to a node, a cusp, a double 
tangent and an inflexion. 
On the general subject, and founded on 374, we have 
Smith, H. J. S., “ On the Higher Singularities of Plane Curves,” Proc. Lond. 
Math. Soc. vol. vi. (1875), pp. 153—182. The author refers to the two following 
enquiries : 
(1) It is important to prove that the indices of singularity as defined by 
Professor Cayley satisfy the equations of Pliicker; and that the “genus” or “deficiency” 
of the plane curve is correctly given by these indices. 
(2) It is also of interest to examine whether any given singularity can be actually 
formed by the coalescence of the ordinary singularities to which it is regarded as 
equivalent : in other words whether a singularity of which the indices are 8, t, k, i 
and which is therefore regarded as equivalent to 8 double points, r double tangents, 
k cusps and i inflexions possesses a penultimate form in which all these singularities 
exist distinct from one another but infinitely close together. 
The paper relates chiefly to the first of these enquiries, the second being reserved 
for a further communication which was never made. 
See also Halphen’s “ Etude sur les points singuliers des courbes algébriques 
planes,” published as an Appendix, pp. 537—648, to the translation of Salmon’s Higher