Contributions to Equilong Geometry
Contributions to Equilong Geometry
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CONTKIBUTIONS TO EQUILONG GEOMETRY. If these equations be solved for the c's, then dh C2 = du" 2! ■ dh ro 21' C3 = du^ 3! dH ~3!' C4 = du'' 4! rz — ro 41 » C5 = dH dw' ' 5! Tz— n 51 » d^'-^v du^-^ p <»/2+l E (- l)^V„+2-2p and ^"+'~ (n+2)! (n + 2)! in which p takes positive integral values. By means of these results, the Cauchy test for the convergency of series may be interpreted geometrically. If the series V = C^U^ + • • • + CkU^ + • • • represent an analytic arc, then the superior limit, as n = oo, of n+2 p < (nl2) + 1 E (- 1)^V2-2P p=l is finite. If this condition be not fulfilled then the series is divergent and represents geometrically not an analytic curve but what Kasner* has called a "divergent differential element of infinite order. " 4. Pedals and Negative Pedals. Suppose the equation of an oriented curve be expressed in polar coordinates (p, 6). Let the coordinates be slightly modified by the following conditions. Let the direction on the radius vector (p) from the pole to a point of a curve be positive or negative according as the pole lies on the left or on the right of the positive direction of the curve at the point.
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To quickly assess the difficulty of the text, read a short excerpt:
CONTKIBUTIONS TO EQUILONG GEOMETRY. If these equations be solved for the c's, then dh C2 = du" 2! ■ dh ro 21' C3 = du^ 3! dH ~3!' C4 = du'' 4! rz — ro 41 » C5 = dH dw' ' 5! Tz— n 51 » d^'-^v du^-^ p <»/2+l E (- l)^V„+2-2p and ^"+'~ (n+2)! (n + 2)! in which p takes positive integral values. By means of these results, the Cauchy test for the convergency of series may be interpreted geometrically. If the series V = C^U^ + • • • + CkU^ + • • • represent an analytic arc, then the superior limit, as n = oo, of n+2 p < (nl2) + 1 E (- 1)^V2-2P p=l is finite. If this condition be not fulfilled then the series is divergent and represents geometrically not an analytic curve but what Kasner* has called a "divergent differential element of infinite order. " 4. Pedals and Negative Pedals. Suppose the equation of an oriented curve be expressed in polar coordinates (p, 6). Let the coordinates be slightly modified by the following conditions. Let the direction on the radius vector (p) from the pole to a point of a curve be positive or negative according as the pole lies on the left or on the right of the positive direction of the curve at the point.
What to read after Contributions to Equilong Geometry?
You can find similar books in the "Read Also" column, or choose other free books by Paul Henry Linehan to read online
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