- For the theory of Steinerian couples and quadruples on plane cubics see the author's Introduction to Projective Geometry, New York, 1905, pp. 197–204.
- See D. E. Smith, Amer. Math. Monthly, Vol. 24, 1917, p. 64.
- Brocard, Now. Corr. Math., Vol. 6, 1880, p. 211; cf. Loria, Ebene Kurven, 1902, p. 608.
- Mémoires de l'Académic Royale des Sciences, 1732.
- E.g., Math. Monthly (ed. J. D. Runkle), 1, 1859, p. 249. There are also more elaborate papers such as “Sur les courbes de poursuite d'un cercle,” by M. L. Dunoyer, Nouv. Annales de Math., 4th series, Vol. 6, 1906, p. 193. [Compare page 91 of this issue.—Editor].
- The equations of the normal and tangent to the dog's path may be written and x cos ω + y sin ω = cos (ω - θ)-ρ
- x sin ω - y cos ω = sin (ω - θ),
- where ρ denotes the distance between the dog and duck. These equations are satisfied by the coordinates of the dog's position and may be regarded as expressing these coordinates in terms of ω, θ and ρ.
- If we differentiate and divide by dθ, remembering that dy = dx tan ω, and dx sec ω = ds = cdθ, we shall get equations which reduce to
- These are the equations derived by Professor Hathaway in his solution of Problem 2801 given on pages 93–97, theta;, ω — θ ρ, and c being the same as his s, θ, r and k.
- Now v = ρ[2 cos (ω — θ) — ρ], and p = sin (ω — θ), and these equations by aid of (1) and (2) lead directly to Mr. Morley's differential equation in p and v.
- Dunoyer (l. c.) also derives equations (1) and (2) as expressing the components of the dog's velocity in the direction of the duck and at right angles to this direction.—Editor.
- C. Runge, Graphical Methods, Columbia University Press, 1912, p. 120.
- For an interesting and elementary account of the graphical treatment of differential equations, see H. Brodetsky, Mathematical Gazette, October, 1919, and January, March, and May, 1920.
The American Mathematical Monthly
Volume 28, 1921 - Issue 2