On the Cardioids Fulfilling Certain Assigned Conditions
On the Cardioids Fulfilling Certain Assigned Conditions
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To quickly assess the difficulty of the text, read a short excerpt:
X, p. 44. ) PROBLEM (b) : The centre, a point and a line given.
Take the origin as centre; z-\-z' = 2 as the given line and /> as the given point.
The intersections of cusp-loci will, as before, give the number of solutions. The cusp-locus for cardioids with centre and touching z-\-z' = 2 is the circular cubic 27kk'(k-\-k')-5ikk'-\-S = 0; the cusp-locus for cardioids with centre and passing through p is the bicircular quartic Sk''k'^-6pp'kk'-^kk\kp'-\-k'p)-py^ = 0.
These two curves have 12 comm...on intersections; i. E. , there should be 12 cardioids satisfying the required conditions. But, since Cardioids Fulfilling Certain Assigned Conditions 17 the two curves have 4 points in common at each of the circular points, there can be, at most, 4 real cardioids. However, there are not always 4 real. Therefore, let us determine the regions of the plane where there are 4, or only 2, or no real cardioids satis- fying the given conditions.
The map equation of any cardioid with centre at is z = ^at-aT, Since 2+2' = 2 is to be tangent to this, z-Sat+Sa'f-z't^ -s+z'-^ Employing the value t= —1, we have for a, a\ any one of the 8 equations : 3a+3a/=-2 3aco+3aV=-2 3aco2+3a'a;=-2 It will be sufficient to take the first equation.
What to read after On the Cardioids Fulfilling Certain Assigned Conditions?
You can find similar books in the "Read Also" column, or choose other free books by Mary Gervase Kelley to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
X, p. 44. ) PROBLEM (b) : The centre, a point and a line given.
Take the origin as centre; z-\-z' = 2 as the given line and /> as the given point.
The intersections of cusp-loci will, as before, give the number of solutions. The cusp-locus for cardioids with centre and touching z-\-z' = 2 is the circular cubic 27kk'(k-\-k')-5ikk'-\-S = 0; the cusp-locus for cardioids with centre and passing through p is the bicircular quartic Sk''k'^-6pp'kk'-^kk\kp'-\-k'p)-py^ = 0.
These two curves have 12 comm...on intersections; i. E. , there should be 12 cardioids satisfying the required conditions. But, since Cardioids Fulfilling Certain Assigned Conditions 17 the two curves have 4 points in common at each of the circular points, there can be, at most, 4 real cardioids. However, there are not always 4 real. Therefore, let us determine the regions of the plane where there are 4, or only 2, or no real cardioids satis- fying the given conditions.
The map equation of any cardioid with centre at is z = ^at-aT, Since 2+2' = 2 is to be tangent to this, z-Sat+Sa'f-z't^ -s+z'-^ Employing the value t= —1, we have for a, a\ any one of the 8 equations : 3a+3a/=-2 3aco+3aV=-2 3aco2+3a'a;=-2 It will be sufficient to take the first equation.
What to read after On the Cardioids Fulfilling Certain Assigned Conditions?
You can find similar books in the "Read Also" column, or choose other free books by Mary Gervase Kelley to read onlineMoreLess
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