. An elementary treatise on the differential calculus founded on the method of rates or fluxions. putting 6 + 1800in place of 6, we have 292 CERTAIN HIGHER PLANE CURVES. [Art. 267. r = a cos 26cos 6 (4) the polar equation of the strophoid, B being the pole and BAthe initial line. Examples. 1. Find the position of the maximum ordinate of the strophoid. The origin being A, x = \ (3 — 4/5) a. 2. Let A be sl fixed point in the circumference of a circle, and ABany chord passing through it; draw a diameter parallel to AB, andthrough B a line parallel to the tangent at A ; prove that the locus ofthe
. An elementary treatise on the differential calculus founded on the method of rates or fluxions. putting 6 + 1800in place of 6, we have 292 CERTAIN HIGHER PLANE CURVES. [Art. 267. r = a cos 26cos 6 (4) the polar equation of the strophoid, B being the pole and BAthe initial line. Examples. 1. Find the position of the maximum ordinate of the strophoid. The origin being A, x = \ (3 — 4/5) a. 2. Let A be sl fixed point in the circumference of a circle, and ABany chord passing through it; draw a diameter parallel to AB, andthrough B a line parallel to the tangent at A ; prove that the locus ofthe intersection of this line with the diameter is the strophoid. The Limacon of Pascal. 268. If through a given point A on the circumference of acircle a line be drawn cutting the circumference again at C, andfrom the latter point a given distance be laid off in each direc-tion on this line, the locus of the points thus determined iscalled the limagon. Let the diameter of the circleACB, Fig. 43, be denoted by 2a,and the given constant by b;then the polar equation of thelocus of P and P will be. r = 2a cos 6 ± b. (1) Fig. the point defined by It is to be observed that eachof these equations gives the en-tire curve; for, if we put /2+1800for 6, and use the lower sign, we §XXXI.| THE LIMACON OF PASCAL. 293 6 = a + 1800, and r = — 2a cos a — b ; but this is the same as thepoint determined by 6 = a, and r — 2a cos <r + b, Ithe latter being obtained by ; using the upper sign in equa-tion (1). Reversing the direction ofthe initial line, we have r = b — 2a cos 6, . (2) another form of the polarequation.
Size: 1665px × 1501px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdecade1870, bookpublishernewyo, bookyear1879
