Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-05T07:47:27.305Z Has data issue: false hasContentIssue false

Expansions of coordinates of points of a plane curve in terms of s or Ψ

Published online by Cambridge University Press:  24 October 2008

H. T. H. Piaggio
Affiliation:
University CollegeNottingham

Extract

If O is an ordinary point of a plane curve and the tangent and inward normal at O are taken as axes, then, under certain conditions, the coordinates x and y of a neighbouring point P on the curve can be expanded in powers of s, the arc OP, or of Ψ, the angle between the tangents at O and P. Lamb(3) gives these expansions as far as the terms in s4 and Ψ3. Dockeray(1) goes as far as the terms in s6, but there seem to be three errors in his results. No one, I believe, has given the general terms. I obtain these for both pairs of expansions. Fowler(2) gives the term in s2r+1 in the expansion of x in the special case where, owing to the vanishing of the curvature and of its first r − 2 derivatives, the expansion of y begins with the term in sr+1. I shall give reasons for disagreeing with this result.

Type
Research Notes
Copyright
Copyright © Cambridge Philosophical Society 1945

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1Dockeray, . Pure Mathematics (London, 1934), p. 395.Google Scholar
2Fowler, . Elementary Differential Geometry of Plane Curves (Cambridge, 1920), p. 44.Google Scholar
3Lamb, . Infinitesimal Calculus, 3rd ed. (Cambridge, 1919), p. 499.Google Scholar
4Mathews, . Quart. J. Math. 26 (1893), 27.Google Scholar