An Elementary Treatise On the Differential Calculus, With Applications And Numerous Examples;
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4. Show that the inverse of a conic with regard to the focus is a Lima- 5on (Equation r=a + 6 cos ff), which becomes a cardioide if the conic be a parabola.
5. Show that the Equation of the inverse of a conic with regard to the centre is of the form r^=A+B coa29, which becomes a Lemniscate of Bernoulli if the conic be a rectangular hyperbola.
174 CHAPTER Vn.
229. Tangents to Curve and Inverse inclined to Radius Tector at Supplementary Angles If P, P' be two contiguous points on a curve, and Q, ...Q' the inverse points, then, since OP . OQ=OP' . OQ', the points P, P', ■Q', Q are coney clic ; and since the angles OPT and OQT are therefore supplementary, it follows that in the limit when P' Fig. 29.
ultimately coincides with P and Q' with Q, the tangents at P and Q make supplementary angles with OPQ.
The ultimate ratio of corresponding elementary arcs, viz., OP OP . OQ k^y "OQ ds_^PP'_jOP ds'—^^QQ'~-^^OQ'" OQ^ r'2 fc2 230. It follows from the preceding article that when two curves intersect, their inverses intersect at the same angle ; and as particular cases, if two curves touch, their inverses touch, and if the original curves cut orthogonally their inverses cut orthogonally.
What to read after An Elementary Treatise On the Differential Calculus, With Applications And Numerous Examples;?
You can find similar books in the "Read Also" column, or choose other free books by Joseph F Edwards to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
4. Show that the inverse of a conic with regard to the focus is a Lima- 5on (Equation r=a + 6 cos ff), which becomes a cardioide if the conic be a parabola.
5. Show that the Equation of the inverse of a conic with regard to the centre is of the form r^=A+B coa29, which becomes a Lemniscate of Bernoulli if the conic be a rectangular hyperbola.
174 CHAPTER Vn.
229. Tangents to Curve and Inverse inclined to Radius Tector at Supplementary Angles If P, P' be two contiguous points on a curve, and Q, ...Q' the inverse points, then, since OP . OQ=OP' . OQ', the points P, P', ■Q', Q are coney clic ; and since the angles OPT and OQT are therefore supplementary, it follows that in the limit when P' Fig. 29.
ultimately coincides with P and Q' with Q, the tangents at P and Q make supplementary angles with OPQ.
The ultimate ratio of corresponding elementary arcs, viz., OP OP . OQ k^y "OQ ds_^PP'_jOP ds'—^^QQ'~-^^OQ'" OQ^ r'2 fc2 230. It follows from the preceding article that when two curves intersect, their inverses intersect at the same angle ; and as particular cases, if two curves touch, their inverses touch, and if the original curves cut orthogonally their inverses cut orthogonally.
What to read after An Elementary Treatise On the Differential Calculus, With Applications And Numerous Examples;?
You can find similar books in the "Read Also" column, or choose other free books by Joseph F Edwards to read onlineMoreLess
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