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PURE AND APPLIED
J. W. L. GLAISHER, Sc.D., F.R.S.,
FELLOW OF TRINITY COLLEGE, CAMBRIDGE.
LONG MANS, GREEN, AND CO.
CONTENTS OF VOL. XLVII.
PURE AND APPLIED MATHEMATICS.
THE OSCNODAL TRANSFORMATION AND
By A. B. BASSET, M.A., F.R.S.
1. THE present paper is a continuation of my former investigations on the singularities of plane curves. References to my papers On certain Singularities of Plane Curves, On the Reciprocation of the Singularities of Plane Curves, and to my Geometry of Surfaces will be denoted by the letters S. C., R.S., and G.S. respectively.
2. In S.C. § 4, I gave a short account of the oscnodal transformation, which I shall write in the form
where u ayẞ, and the accents are omitted in (2).
Let us denote the primitive curve (1) and the transformed curve (2) by S and respectively. Then if S is arbitrarily situated with respect to the triangle of reference, Σ has a hypertacnode at A, tangent AB, which consists of n branches each of which osculates every other branch, and is therefore formed by the union of three non-collinear multiple points of the first kind and of orders n, which we shall call 1, 2, 3; but when S is specially situated with respect to the triangle of reference, possesses a singularity of the hypertacnode type in which the point 1 is a multiple point of order n of a different character, whilst the points 2 and 3 are of the first kind.
* Quar. Jour. of Math. vol. xliii., p. 151.
† Ibid. vol. xlv., p. 52.