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The character of the point 1 can be examined by transforming (2) by the double quadric transformation

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which gets rid of the points 2 and 3 and converts (2) into a third curve having the special point 1 at some point A' on AC.

The result of this transformation is as follows. Let

v=a (y―a) + By,

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then III. transforms u into ß'v, and u into ẞ"U, where U is the same binary quantic of (a, B) as u is of (B, y). Hence (2) transforms into a third curve

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v" + ẞv"~1 U1 + B3v"' U,+...B" U=0.........(3),

n

2

also if any quantity such as u in (2) is of the form_y'u...”, rs, the corresponding term in (3) will be 'Up' Now let the triangle of reference be changed to A'BC, where y—a=0 is the equation of BA', and write ya+y', then the U's are unchanged, but v becomes

v = a(8+ y) +By'..............

(4). Equations (3) and (4) enable the singularity at A' to be examined, which determines the character of the special point 1.

The point constituents d', x' of the special point 1 must now be found; and since all hypertacnodes of this character are self-reciprocal*, the point and line constituents of the singularity in question are

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The equations of the different branches are usually of the

form

μ

y = x2+λx',

where > 2, and their graphs are of three kinds. When s=q/r and λ= μN,, where , denotes all the 7th roots of unity and is real, the graph consists (i) of a rhamphoid cusp when is even; (ii) of a single real branch when r is odd; (iii) when A is real and s is an integer> 2, the branch is an ordinary one having contacts of different orders with the singular branches.

*In the latter part of this paper certain hypertacnodes will be considered, to which this transforination does not apply, which are not self-reciprocal.

3. In R.S. § 11, I proved that all singularities given by the equation

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where q2 are self-reciprocal. If q> 3r a single application of the quadric transformation (see S.C. p. 152)

α

p' (B' + y)

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converts it into one at Bin which the index of the first term of the series is (9 − r)/r, which is > 2. But if q<3r, then (gr)/2, and the index of the first term of the series is <2 and the transformed singularity is not self-reciprocal. In this case the term must be omitted, although it may possibly appear subsequently as one of the terms of the series. This may be easily proved by starting with the equation

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and proceeding as in R.S. § 11. We thus obtain the following theorem:

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If p, q, r are positive integers where pr <q <(p+ 1) r and P2, the singularity (5) is composed of p non-collinear multiple points of order r, p-1 of which are of the first kind and the remaining one of a different character.

The reader can easily verify this theorem in the case of tacnodes and cusps of different species.

4. The success of the oscnodal transformation depends. upon the primitive curve having some given singularity at B, and I shall now discuss six special cases in which the primitive curve is the cubic

a3 + a3μ‚ + a«‚ + u2= 0......................... and the heading in italics describes the point 1.

(i) Triple point of the second kind.
The equation of the transformed curve is

.(6),

u3 + u3yu, + uy3v ̧ + y *v, = 0.............................. (7), which shows that in the primitive cubic we must write u,=yv1, u ̧=Ÿv,. Hence the cubic touches BA at B. The hypertacnode consists of a cusp of the third species and an

ordinary branch which osculates both the cuspidal branches. The constituents and graphs are given by the equations

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u* + pu3y* + uy3v, + y * v2 = 0 ............................. (8). Hence the cubic has a point of inflexion at B and BA is the tangent thereat. Also

p

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where denotes all the pth roots of unity. The graph consequently consists of a single real branch. By §3, it follows that a single application of the transformation II. converts the index of the second term into 7, which is the singularity formed by the union of a triple point of the first and third kinds; whilst a second application leads to the equation

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which is that of a triple point of the third kind.

The generalisation of this theorem is as follows:

(a) If a curve of the nth degree has n-tactic contact with AB at B, the transformed curve Σ has a hypertacnode formed by the union of two multiple points of the first kind of order n, and another of the same order, the tangents at which coincide.

The constituents of the special point are 8=(n-1) (n−2), x = -1; hence those of the hypertacnode are

d = T = 1 (n − 1) (n − 2) + n (n − 1)

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and the equation of the latter is

y = x2 + λN„ x(3n+1)/n ̧

If n is even, has the two real values +1, so that the graph is a rhamphoid cusp; but if n is odd, 2, has only one real value equal to +1, so that the graph consists of a single

real branch.

The first case furnishes a general theorem which includes the one just considered:

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(3) If a curve of the nth degree has p-tactic contact with AB at B, np, the transformed curve has a hypertacnode

at A formed by the union of two multiple points of the first kind of order n, and a third one of the same order, p of whose tangents coincide, whilst the remaining n-p are distinct.

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y = x2 + λ Q„x (3p+1)'p,

p2, and n-p real ones of the form

y=x+μα.

(iii) A pair of tacnodal branches and one distinct ordinary branch.

The equation is

=

u3 + u2yu,+uy3v, + y3 w1 = 0.........................(9). Hence the cubic has a node at B, nodal tangents arbitrary. Since the special point 1 is formed by the union of a triple point of the first kind and a node, the hypertacnode consists of three ordinary branches two of which have quadritactic contact with one another, whilst the third osculates the other two. Its constituents are 8 = T = 10.

If the primitive curve has a multiple point of the first kind of order p at B, whose tangents are arbitrary and pn-1, it can be shown in the same manner that point 1 is the singularity formed by the union of two multiple points of orders n and p, and therefore consists of p branches which have bitactic contact with one another and n-p distinct ordinary branches. Hence:

The transformed curve has a hypertacnode at A consisting of p branches which have quadritactic contact with one another, and n-p which have tritactic contact with one another and with the remaining p branches.

(iv) The same when the ordinary branch is analytically coincident.

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Hence the cubic has a node at B and BA is one of the nodal tangents.

The point 1 is the special triple point which occurs on the quintic

a2y3 + ay", + " ̧=0.......

..(11),

which belongs to a class of multiple points which I have elsewhere discussed. The point constituents of the singularity are 83, x = 1; hence those of the hypertacnode are 8=T=9, K=6=1.

Now if (11) be transformed by II., the transformed curve will have a node at B, one of whose tangents is BC, whilst the other is arbitrary; hence the graph of the triple point is a cusp with a branch through it which touches the cuspidal tangent. Accordingly, the graph of the hypertacnode consists of a cusp of the third species and an ordinary branch which has quadritactic contact with both the cuspidal branches.

If, however, we had employed a quartic instead of a cubic point 1 would be a quadruple point consisting of the special triple point on (11) with an ordinary branch through it; and this gives rise to a fourth branch having tritactic contact with the other branches.

Hence the generalised theorem is as follows:

(a) Let a curve of the nth degree have a multiple point of the first kind of order p at B, where pn-1, one of whose tangents is BA, whilst the rest are arbitrarily situated; then the transformed curve has a hypertacnode at A consisting of (i) a cusp of the third species, (ii) p-1 ordinary branches which have quadritactic contact with the cuspidal branches and with one another, (iii) n-p-1 ordinary branches which have tritactic contact with themselves and with the remaining branches.

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When n=4 and p=2, the hypertacnode is the limiting form of the curve shown in the figure. The arrows indicate the cuspidal branches, the stars the ordinary branch which

*Quar. Jour. of Math., vol. xxxvii., p. 313; G.S., chap. iv.

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