An Elementary Treatise On the Differential And Integral Calculus With Examples
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Definition of Envelope. The intersection of any two curves of a series will approach a certain limit, as the two curves approach coincidence. Now, if we suppose the param- eter to vary by infinitesimal increments, the locus of the ulti- mate intersections of consecutive curves is called the envelope of the series.
138 DIFFERENTIAL CALCULUS.
133. The envelope of a series of carves is tangent to every curve of the series.
P Q Suppose L, M, N to be any three curves of the series. P is the intersec...tion of M with the preceding curve L, and Q its intersection with the following curve N.
As the curves approach coincidence, P and Q will ultimately be two consecutive points of the envelope, and of the curve M. Hence the envelope touches M.
Similarly, it may be shown that the envelope touches any other curve of the series.
134. To find the equation of the envelope of a given series of curves.
Before considering the gen- eral problem let us take the following special example.
Eequired the envelope of the series of straight lines represented by y = ax + ™, a a being the variable param- eter.
What to read after An Elementary Treatise On the Differential And Integral Calculus With Examples?
You can find similar books in the "Read Also" column, or choose other free books by George a George Abbott Osborne to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Definition of Envelope. The intersection of any two curves of a series will approach a certain limit, as the two curves approach coincidence. Now, if we suppose the param- eter to vary by infinitesimal increments, the locus of the ulti- mate intersections of consecutive curves is called the envelope of the series.
138 DIFFERENTIAL CALCULUS.
133. The envelope of a series of carves is tangent to every curve of the series.
P Q Suppose L, M, N to be any three curves of the series. P is the intersec...tion of M with the preceding curve L, and Q its intersection with the following curve N.
As the curves approach coincidence, P and Q will ultimately be two consecutive points of the envelope, and of the curve M. Hence the envelope touches M.
Similarly, it may be shown that the envelope touches any other curve of the series.
134. To find the equation of the envelope of a given series of curves.
Before considering the gen- eral problem let us take the following special example.
Eequired the envelope of the series of straight lines represented by y = ax + ™, a a being the variable param- eter.
What to read after An Elementary Treatise On the Differential And Integral Calculus With Examples?
You can find similar books in the "Read Also" column, or choose other free books by George a George Abbott Osborne to read onlineMoreLess
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