An elementary course of infinitesimal calculus . Fig. 105. If the constant of inversion be changed we get a similar curve,which will still be a circle through the centre of similitude 0. 380 INFINITESIMAL CALCULUS. [CH. IX Ex. 2. More generally, the inverse of any circle is a Fig. 106. Let 0 be the centre of inversion, C the centre of the givencircle, a its radius; and let le = OG--a (6). If, then, we draw any chord OPP through 0, it is known fromGeometry that OP. OP= 00-a?^k (7) Hence P traces out the inverse of the locus of P; the circleinverts into itself. And by changing the
An elementary course of infinitesimal calculus . Fig. 105. If the constant of inversion be changed we get a similar curve,which will still be a circle through the centre of similitude 0. 380 INFINITESIMAL CALCULUS. [CH. IX Ex. 2. More generally, the inverse of any circle is a Fig. 106. Let 0 be the centre of inversion, C the centre of the givencircle, a its radius; and let le = OG--a (6). If, then, we draw any chord OPP through 0, it is known fromGeometry that OP. OP= 00-a?^k (7) Hence P traces out the inverse of the locus of P; the circleinverts into itself. And by changing the constant of inversion we get a similarcurve, and therefore a circle. If, as in the right-hand figure, 0 be within the given circle,the constant in (7) is negative. This means that P and P arenow on opposite sides of 0. 146. Mechanical Inversion. There are various devices by which the inverse of a givencurve can be traced mechanically. 1°. Peaucelliers Linkage. This consists of a rhombus PAQB formed of four rodsfreely jointed at their extremities, and of two equal bars con-necting two opposite corners A,B to a, fixed pivot at 0. It is evident that, whatever shape and position thelinkage assumes, the points P, Q will always be in a straight 145-146] SPECIAL CURVES. 381 line wit
Size: 2511px × 995px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1900, bookpublishercambr, bookyear1902
