. Differential and integral calculus. em are said to be of thesame family. Thus the equation (x — a)2 -f- (y—b)2 = ^2 is the equation of afamily of a circles whose positions and magnitudes depend uponthe values of the constants a, b, r. Again, the equation Ax +By + C = o is the equation of a family of straight lines whosedirections and positions with reference to the axes depend uponthe values of A, B, C. The constants which enter equations are called Parameters,and if one or more of these are supposed to vary, they arecalled Variable Parameters. 146. Envelope. The Envelope of a family of curv
. Differential and integral calculus. em are said to be of thesame family. Thus the equation (x — a)2 -f- (y—b)2 = ^2 is the equation of afamily of a circles whose positions and magnitudes depend uponthe values of the constants a, b, r. Again, the equation Ax +By + C = o is the equation of a family of straight lines whosedirections and positions with reference to the axes depend uponthe values of A, B, C. The constants which enter equations are called Parameters,and if one or more of these are supposed to vary, they arecalled Variable Parameters. 146. Envelope. The Envelope of a family of curves is a curvetangent to each member of the family. Thus, if we assume a to be the variable parameter in theequation (x — a)2 -f- (y — b)2= r2, b and r remaining constant,we have (Fig. 31) a series of circles, all of whose centers are onthe line MN, at a distance b from the .r-axis. The envelopesof this family of circles are evidently the || lines AB and EFtwhose equations are y = b ± r. Contact of Curves Envelopes 201 we and. If we assume b to vary, a and r remaining constant, we havea family of circles whose centers lie along the line LK at a dis-tance a from the envelopes in thiscase being CD andHG, whose equations are x = a ± r. Similarly,suppose a vary, r remaining con-stant, or a and r tovary, b remaining con-stant, etc.; or we maysuppose all three tovary at the same each case we havea family of circles, and a curve tangent to the members of thatfamily is called the envelope. It is evident, in this case, that if r alone varies there is noenvelope. 147. To determine the equation of the envelope. Let u =/(x, y, a) = o . . (V) be the equation of any one (MN) of a family of curves,a being the variable param-eter, and let u =f(x,y) =o . . (d) be the equation ofthe envelope ST. Let P(x, y) be the point of tan-gency. Since the curves are tan-gent to each other at (x, y) the first derivatives drawn from their equations must be equal; hence differentiat
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1918
