A Treatise on the Higher Plane Curves: Intended as a Sequel to A Treatise on Conic Sections
Hodges and Smith, 1852 - Conic sections - 316 pages
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Common terms and phrases
angle appears apply asymptotes axis becomes branches centre circle co-ordinates coefficients coincide common condition conic consecutive considered constant contain corresponding cubic cusp denotes determine distances double point drawn elimination ellipse envelope equal equation evolute example expressed figure fixed points foci four function give given given point Hence imaginary infinite infinity intersection line joining locus manner meets the curve method multiple point normal nth degree obtained origin oval pair parallel pass perpendicular points of contact points of inflexion polar polar conic pole position problem properties proved ratio reader reciprocal reduced regard relation represents result right line roots seen sides substituting tangent theorem third degree tion touch transformation triangle values variable
Page 5 - The line joining the middle points of two sides of a triangle is parallel to the third side and equal to half of the third side.
Page 241 - ... the locus of the centres of circles described through the origin to touch the inverse curve. Thus from the theorem that the locus of the foot of the perpendicular from the focus on the tangent of a conic is a circle, we deduce (as Mr.
Page 166 - The facility which this mode of expressing points on a cubic of the third class, gives to the management of questions concerning them, renders them nearly as easy to be treated as conic sections. Many theorems, too, are thus suggested, which, though only thus proved for curves of the third class, if they do not involve any mention of the cusp or of the number of tangents which can be drawn to the curve from any point, we may see will be true for all curves of the third degree. Ex. 1. A chord is drawn...
Page 225 - Patris veluti imago, et ab illo ut lumen a lumine emanans, eidem ор.оаиа-ч>; existit qualiscunque adumbratio. Aut, si mavis, quia curva nostra mirabilis in ipsa mutatione semper sibi constantissime manet similis et numero eadem, poterit esse, vel fortitudinis et constan ti;i...
Page 206 - This curve is generated by the motion of a point on the circumference of a circle which rolls along a right (line.
Page 204 - ... a curve having the origin for a double point, and the two circular points at infinity for ordinary double points. As a generalization of the ovals of Cassini, we might seek the locus of a point, the product of whose distances from m given points shall be constant ; and...
Page 118 - PRA, the angle which the incident ray makes with the normal to the surface...
Page 241 - Generally the reciprocal of a curve is the locus of the centres of circles through the origin to touch the inverse curve. Thus, " the envelope of a circle passing through a given point, and having its centre on a parabola, is a circular cubic with a double point: and having its centre on any conic, is a bicircular curve of the fourth degree having a finite double point.
Page 203 - From this show that the curve has three foci ; te three evanescent circles having double contact with the curve. 8. The base angles of a variable triangle move on two fixed circles, while the two sides pass through the centres of the circles, and the base passes through a fixed point on the line joining the centres ; prove that the locus of the vertex is a Cartesian. 9. Prove that the inverse of a Cartesian with respect to any point is a bicircular quartic.
Page 136 - B 3 = 0 , an equation also containing nine constants. A, C, E are (Art. 42) tangents at points of inflexion, which lie on a right line B, and we have the theorem proved already, Art. 49, the line joining two points of inflexion must pass through a third. And again, the cube of the distance of any point of the curve from this line is in a constant ratio to the product of its distances from the three tangents at the points of inflexion. We give here another form, on account of its connexion with points...