DESCRIPTION OP OVAL CURVES. 3 



tn 



within it ; and in this case the ratio expresses the index of refraction of 



tin 



the medium*. 



If we denote by the power of either focus the number of strings leading 

 to it by Mr Maxwell's construction, and if one of the foci be removed to an 

 infinite distance, if the powers of the two foci be equal the curve is a parabola; 

 if the power of the nearer focus be greater than the other, the curve is an 

 ellipse ; if the power of the infinitely distant focus be the greater, the curve 

 is a hyperbola. The first case evidently corresponds to the case of the reflection 

 of parallel rays to a focus, the velocity being unchanged after reflection ; the 

 second, to the refraction of parallel rays to a focus in a dense medium (in 

 which light moves slower) ; the third case to refraction into a rarer medium. 



The ovals of Descartes were described in his Geometry, where he has also 

 given a mechanical method of describing one of themf, but only in a particular 

 case, and the method is less simple than Mr Maxwell's. The demonstration of 

 the optical properties was given by Newton in the Principia, Book i., prop. 97, 

 by the law of the sines; and by Huyghens in 1690, on the Theory of Undu- 

 lations in his Traite de la Lumiere. It probably has not been suspected that 

 so easy and elegant a method exists of describing these curves by the use of 

 a thread and pins whenever the powers of the foci are commensurable. For 

 instance, the curve, Fig. 2, drawn with powers 3 and 2 respectively, give the 

 proper form for a refracting surface of a glass, whose index of refraction is 1'50, 

 in order that rays diverging from f may be refracted to F. 



As to the higher classes of curves with three or more focal points, we 

 cannot at present invest them with equally clear and curious physical properties, 

 but the method of drawing a curve by so simple a contrivance, which shall 

 satisfy the condition 



mr + nr +pr" + &c. = constant, 



is in itself not a little interesting; and if we regard, with Mr Maxwell, the 

 ovala above described, as the limiting case of the others by the coalescence 

 of two or more foci, we have a farther generalization of the same kind as that 

 so highly recommended by Montucla|, by which Descartes elucidated the conic 

 sections as particular cases of his oval curves. 



* This was perfectly well shewn by Huyghens in his Traite de la Lumiere, p. 111. (1690.) 



t Edit. 1683. Geometria, Lib. n. p. 54. 



J Histoire des Mathematlques. First Edit. n. 102. 



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