An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page 6
... expression for r2 . It may also be proved in a similar manner that = abc 442 { a cos A ( α , -a , ) 2 + b cos B ( ß ̧ — ß ‚ ) 2 + ccos C ' ( y1 − y2 ) 2 } . 4. We next proceed to investigate the equation of a straight line ; and first ...
... expression for r2 . It may also be proved in a similar manner that = abc 442 { a cos A ( α , -a , ) 2 + b cos B ( ß ̧ — ß ‚ ) 2 + ccos C ' ( y1 − y2 ) 2 } . 4. We next proceed to investigate the equation of a straight line ; and first ...
Page 16
... expression of the fact that the two equations la + mB + ny = 0 , l'a + m'ß + n'y = 0 , cannot be simultaneously satisfied by any values of a , B , y , or , in other words , that the two straight lines represented by them do not ...
... expression of the fact that the two equations la + mB + ny = 0 , l'a + m'ß + n'y = 0 , cannot be simultaneously satisfied by any values of a , B , y , or , in other words , that the two straight lines represented by them do not ...
Page 17
... expressed by saying that the equation ax + bB + cy = 0 , or the equivalent equation a sin A + B sin B + y sin C = 0 , represents the straight line at infinity . This phraseology is very convenient , and free from objec- tion , if the ...
... expressed by saying that the equation ax + bB + cy = 0 , or the equivalent equation a sin A + B sin B + y sin C = 0 , represents the straight line at infinity . This phraseology is very convenient , and free from objec- tion , if the ...
Page 21
... expression . It will be observed , that the numerator of this expression vanishes if the point ( f , g , h ) lie upon the line ( l , m , n ) , as manifestly ought to be the case . It will also be remarked , that the more nearly the ...
... expression . It will be observed , that the numerator of this expression vanishes if the point ( f , g , h ) lie upon the line ( l , m , n ) , as manifestly ought to be the case . It will also be remarked , that the more nearly the ...
Page 23
... expressed by the notation { O. PQRS } * . DEF . 2. If P , Q , R , S be four points in a straight line , PQ . RS the ratio is called the anharmonic ratio of the range PS . QR P , Q , R , S , and may be expressed thus [ PQRS ] . in using ...
... expressed by the notation { O. PQRS } * . DEF . 2. If P , Q , R , S be four points in a straight line , PQ . RS the ratio is called the anharmonic ratio of the range PS . QR P , Q , R , S , and may be expressed thus [ PQRS ] . in using ...
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angular points anharmonic ratio asymptotes auxiliary conic b₁ b₂ C₁ centre Chap chord co-ordinates coefficients common tangents conic section determinant directrix find the equation fixed point fixed straight line follows four points given conic given point given straight line given triangle Hence investigated Let the equation line at infinity line joining locus meets the conic nine-point circle obtain opposite sides parabola Pascal's Theorem perpendicular point f points at infinity points of contact points of intersection PQRS prove radical axis reciprocal curve reciprocated with respect rectangular hyperbola represented right angles second degree shewn similar and similarly sin POS tangents drawn term a,b,c theorem three points three straight lines triangle of reference ua² uf+w'g+v'h ux² V'ca va² Vb² vß² W'ab whence wy² λα