On the Pentacardioid
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We thus obtain, as the equations of the three cusps, 3ci = — 6^1 + 2tl/t^ — 2t% + t{, ' Sc^ = — 6<2 + 2tl/t, — 2tlt, + 4, (39) 3c3 = — 6^3 + 2tyti — 2tlt, + tl . This of course is not the only form in which the equations can be put ; and by replacing one of the parameters by its value derived from the involution Sg + 1 = 0, we get another form which will be found useful to us, namely 3c3 = 6/t, t, + 2/tltl + 2lt\tl + \lt\tl, (40) and two similar equations obtained by cyclically interchanging the subscripts. Now let us hold one of the parameters, t-^ say, fixed and allow the other two to vary, replacing them by t. Under these conditions the three cusps will have to move along definite curves, since their positions depend on the single variable t. The cusp Ci is by this means separated, as it were, from the other two, and these latter lose their identity and may be treated as being a pair interchangeable at will. The cusp Cj moves along one curve, namely 3ci = — 6
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To quickly assess the difficulty of the text, read a short excerpt:
We thus obtain, as the equations of the three cusps, 3ci = — 6^1 + 2tl/t^ — 2t% + t{, ' Sc^ = — 6<2 + 2tl/t, — 2tlt, + 4, (39) 3c3 = — 6^3 + 2tyti — 2tlt, + tl . This of course is not the only form in which the equations can be put ; and by replacing one of the parameters by its value derived from the involution Sg + 1 = 0, we get another form which will be found useful to us, namely 3c3 = 6/t, t, + 2/tltl + 2lt\tl + \lt\tl, (40) and two similar equations obtained by cyclically interchanging the subscripts. Now let us hold one of the parameters, t-^ say, fixed and allow the other two to vary, replacing them by t. Under these conditions the three cusps will have to move along definite curves, since their positions depend on the single variable t. The cusp Ci is by this means separated, as it were, from the other two, and these latter lose their identity and may be treated as being a pair interchangeable at will. The cusp Cj moves along one curve, namely 3ci = — 6
What to read after On the Pentacardioid?
You can find similar books in the "Read Also" column, or choose other free books by Edward Charles Phillips to read online
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