Theory of Differential Equations volume 5
Theory of Differential Equations volume 5
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In our eReader you can find the full English version of the book. Read Theory of Differential Equations volume 5 Online - link to read the book on full screen. Our eReader also allows you to upload and read Pdf, Txt, ePub and fb2 books. In the Mini eReder on the page below you can quickly view all pages of the book - Read Book Theory of Differential Equations volume 5
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The equation of a sphere, that is absolutely unrestricted, contains four arbitrary independent constants : hence, when it is made subject to two independent conditions, the equation will contain two arbitrary constants and may be regarded as giving the complete integral of some partial diffe rential equation.
To obtain a general integral, we select a family of these spheres and construct their envelope. The characteristics, being the intersections of consecutive spheres, are circles : each sphe...re touches the envelope surface along a circle. The edge of regression on the envelope surface, which is the general integral, is itself the envelope of the characteristic circles ; and the earlier investigation shewed that, where the sphere meets this edge of regression, it meets the edge in three consecutive points.
But there is one general integral for which the contact is closer. By an appropriate choice of a functional relation between the two constants in the complete integral, we obtain the envelope of one selected family of spheres : each sphere, where it meets the edge of regression on this envelope surface, meets it in four consecutive points, and therefore is its osculating sphere.
What to read after Theory of Differential Equations volume 5?
You can find similar books in the "Read Also" column, or choose other free books by Forsyth, Andrew Russell, 1858-1942 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
The equation of a sphere, that is absolutely unrestricted, contains four arbitrary independent constants : hence, when it is made subject to two independent conditions, the equation will contain two arbitrary constants and may be regarded as giving the complete integral of some partial diffe rential equation.
To obtain a general integral, we select a family of these spheres and construct their envelope. The characteristics, being the intersections of consecutive spheres, are circles : each sphe...re touches the envelope surface along a circle. The edge of regression on the envelope surface, which is the general integral, is itself the envelope of the characteristic circles ; and the earlier investigation shewed that, where the sphere meets this edge of regression, it meets the edge in three consecutive points.
But there is one general integral for which the contact is closer. By an appropriate choice of a functional relation between the two constants in the complete integral, we obtain the envelope of one selected family of spheres : each sphere, where it meets the edge of regression on this envelope surface, meets it in four consecutive points, and therefore is its osculating sphere.
What to read after Theory of Differential Equations volume 5?
You can find similar books in the "Read Also" column, or choose other free books by Forsyth, Andrew Russell, 1858-1942 to read onlineMoreLess
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