Advanced calculus; . tion F(x, y,a) = 0be considered as representing a family of curves where the differentcurves of the family are obtained by assigning different values to theparameter a. Such families are illustrated by (x — af -f- if = 1 and ax -f- y/a = 1, (33) which are circles of unit radius centered on the sc-axis and lines whichcut off the area \ a2 from the first quadrant. As a changes, the circlesremain always tangent to the two lines i/=±l andthe point of tangency traces those lines. Again, as Ya changes, the lines (33) remain tangent to the hyper-bola xy = 7x, owing to the propert
Advanced calculus; . tion F(x, y,a) = 0be considered as representing a family of curves where the differentcurves of the family are obtained by assigning different values to theparameter a. Such families are illustrated by (x — af -f- if = 1 and ax -f- y/a = 1, (33) which are circles of unit radius centered on the sc-axis and lines whichcut off the area \ a2 from the first quadrant. As a changes, the circlesremain always tangent to the two lines i/=±l andthe point of tangency traces those lines. Again, as Ya changes, the lines (33) remain tangent to the hyper-bola xy = 7x, owing to the property of the hyperbolathat a tangent forms a triangle of constant area withthe asymptotes. The lines y = ± 1 are called the -envelope of the system of circles and the hyperbolaxy — k the envelope of the set of lines. In general, if there is a curveto which the curves of a family F(x, y, a) = 0 are tangent and if thepjoint of tangency describes that curve as a varies, the curve is called * = jlxi±\^j(h±\ andflnd *. 136 DIFFERENTIAL CALCULUS the envelope (or part of the envelope if there are several such curves)of the family F(x, y, a) = 0. Thus any curve may be regarded as theenvelope of its tangents or as the envelope of its circles of curvature. To find the equations of the envelope note that by definition theenveloping curves of the family F(x, y,a) = 0 are tangent to the envelopeand that the point of tangency moves along the envelope as a equation of the envelope may therefore be written x = 4>(a), y = if/(a) with F(, if/, a) = 0, (34) where the first equations express the dependence of the points on theenvelope upon the parameter a and the last equation states that eachpoint of the envelope lies also on some curve of the family F(x, y, a) = (34) with respect to a. Then Fx\a) + FtfXa) + Fa ~ 0. (35) Now if the point of contact of the envelope with the curve F = 0 is anordinary point of that curve, the tangent to the curve is Fx(x-x0) + F^y-y
Size: 1539px × 1624px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, bookpublisherbostonnewyorketcgi
