Solutions of the Cambridge Senate House Problems for Four Years 1848 1851
Solutions of the Cambridge Senate House Problems for Four Years 1848 1851
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Prove that the tangent plane at any point of the surface [axY + iJyyY + [czY = 2 {bcyz + cazx + ahxy)^ intersects the surface ayz + hzx + cxy — in two straight lines at right angles to one another.
1850. ] GEOMETRY OF THREE DIMENSIONS. 21. 3 The equation {axY + [hjY + {cz)'' = 2 [hcyz + cazx + ahxy) may be put into the fomi {ax)^ + [hyY- + [czY- = (1).
Also ayz + hzx + cxy = may be wiitten « ^ c „, _.
- + -4 - = 2 .
x y z The equation to the tangent plane to (1) at any point [xyz] is Let [l^m^n...^^ iO^h^^) ^® ^^^ direction-cosines of the lines m which (2) meets (3), then the condition of these being at right angles to one another, is IJ^ + m^m.^ + n^n^ = (4).
Now where (2) meets (3), we have, writing x^y^z^ for xyz in (2), «(ir-(r-(ir-(-)'i(iy^(i)'a' a quadratic m -- whose roots are — ^, — ^ . Hence ©'^. -(D'^'-e/^-" (^'- njWjj VC2; similarly -^-^ = f — ) : n^n^ \czj ' . -. IJ^ + ?n, ?«, + », ?i, =c (aa-)* + (%)* + [czf = by (1).
Hence the tangent plane at any point of (1) cuts (3) in two straight lines at right angles to one another.
What to read after Solutions of the Cambridge Senate House Problems for Four Years 1848 1851?
You can find similar books in the "Read Also" column, or choose other free books by Norman Macleod Ferrers to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Prove that the tangent plane at any point of the surface [axY + iJyyY + [czY = 2 {bcyz + cazx + ahxy)^ intersects the surface ayz + hzx + cxy — in two straight lines at right angles to one another.
1850. ] GEOMETRY OF THREE DIMENSIONS. 21. 3 The equation {axY + [hjY + {cz)'' = 2 [hcyz + cazx + ahxy) may be put into the fomi {ax)^ + [hyY- + [czY- = (1).
Also ayz + hzx + cxy = may be wiitten « ^ c „, _.
- + -4 - = 2 .
x y z The equation to the tangent plane to (1) at any point [xyz] is Let [l^m^n...^^ iO^h^^) ^® ^^^ direction-cosines of the lines m which (2) meets (3), then the condition of these being at right angles to one another, is IJ^ + m^m.^ + n^n^ = (4).
Now where (2) meets (3), we have, writing x^y^z^ for xyz in (2), «(ir-(r-(ir-(-)'i(iy^(i)'a' a quadratic m -- whose roots are — ^, — ^ . Hence ©'^. -(D'^'-e/^-" (^'- njWjj VC2; similarly -^-^ = f — ) : n^n^ \czj ' . -. IJ^ + ?n, ?«, + », ?i, =c (aa-)* + (%)* + [czf = by (1).
Hence the tangent plane at any point of (1) cuts (3) in two straight lines at right angles to one another.
What to read after Solutions of the Cambridge Senate House Problems for Four Years 1848 1851?
You can find similar books in the "Read Also" column, or choose other free books by Norman Macleod Ferrers to read onlineMoreLess
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