. Differential and integral calculus. Contact of Curves Envelopes 205 2 a ^WJ .-. 27 ay2 = 4 (x — 2 #)8, which is the desired equation. We have previously deducedthis equation by the direct method. See §138 and figure 28. 4. The hypotenuse of a righttriangle changes its position,its length remaining unaltered;find its envelope. Let OB A be the triangle,BA being any one position ofthe hypotenuse. Let BA = c, a constant, OB= b, OA = a. Then the equa-tion of BA is. Fig. 33. and by condition, * + { = * a 0 (») Let us take # as the variable parameter. Ordinarily wewould find the value of b in terms
. Differential and integral calculus. Contact of Curves Envelopes 205 2 a ^WJ .-. 27 ay2 = 4 (x — 2 #)8, which is the desired equation. We have previously deducedthis equation by the direct method. See §138 and figure 28. 4. The hypotenuse of a righttriangle changes its position,its length remaining unaltered;find its envelope. Let OB A be the triangle,BA being any one position ofthe hypotenuse. Let BA = c, a constant, OB= b, OA = a. Then the equa-tion of BA is. Fig. 33. and by condition, * + { = * a 0 (») Let us take # as the variable parameter. Ordinarily wewould find the value of b in terms of a from the given condi-tion, and substitute in the equation of the line, and then pro-ceed as in preceding examples; but in this case, as in otherswith which we have had to deal, the simpler process is to sub-stitute after differentiation. Since b is a function of a, we have from (m)> du _ x y dbda cP & da 0») 206 Differential Calculus from (n), 2a -f 2 b— = o; ^ _ tf dfo :~~ V This value in x* -|- y* = <rs for the equation of the envelope. This curve is the four-cuspedhypocycloid, and is generated by a point on the circumferenceof a circle as it rolls on the concave side of another circle whosediameter is four times that of t
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1918
