266 
ON THE CUSP OF THE SECOND KIND OR NODECUSP. 
[343 
this generation of it, be termed, a “nodecusp.” It is to be noticed that in the point- 
theory of curves, there is between the cusp and the nodecusp the intermediate singularity 
of the tacnode, which arises from the union and amalgamation of two nodes, and 
possesses the character of a cusp. 
I return to the nodecusp; taking the point in question as the origin, and the 
tangent for the axis of x, the equation will be a specialised form of the equation 
i + i («> b, c, d#x, y) 3 + fa (a', V, c', d', e'][x, y) 4 + &c. = 0, 
which belongs to the case of a cusp, viz. (see Pliicker, p. 165) the conditions satisfied 
by the special form are a — 0, a! — 3b 2 , or the equation is 
i (y + \ bx 2 f + a y 2 (3cx + dy) + fa y (4b’x? + 6c'x 2 y + 4d'xy 2 + e'y 3 ) + &c. = 0, 
which is most easily verified, by observing that (this being so) the expansion of y in 
terms of x will be of the form 
y — — \ bx 2 + Ax^ + &c. 
It is now to be shown how the foregoing conditions a = 0, a’ = 3b 2 , are obtained by 
assuming that the curve has, besides the cusp, a node which ultimately coincides with 
the cusp. Let (a, /3) be the coordinates of the node; we must have 
2 ft 1 + £ (a, b, c, dQa, /3) 3 + fa (a', b', c', d', e'Qa, /3) 4 4- &c. = 0, 
\ (a, b, c$ a, /3) 2 + £ (a', b', c', d'^a, A) 3 + &c. = 0, 
A + \ (b, c, /3) 2 + ^ (b', c', d!, e\a, /3) 3 + &c. = 0. 
Assume /3 = mar, and then let a vanish; the equations become in the first instance 
^aa 3 + (^m 2 + ^bm + fa a) a 4 + &c. = 0, 
\ aa 2 + &c. = 0, 
(m +b) a 2 + &c. = 0, 
the second and third equation give a = 0, m + £ b = 0, and the first equation then gives 
m 2 + \ bm + fa cl' = 0; 
or substituting for m its value = — \b, this is a' = 3b 2 , or the required conditions are 
a = 0, a' = 36 2 , ut suprct. The single condition a = 0 corresponds to the case of the 
tacnode. 
2, Stone Buildings, W.G., September 17, 1862.