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Differential geometry and its applications: proceedings of the 10th International Conference, DGA 2007, Olomouc, Czech Republic, 27-31 August 2007

This quantity includes invited lectures and chosen examine papers within the fields of classical and sleek differential geometry, worldwide research, and geometric equipment in physics, provided on the tenth overseas convention on Differential Geometry and its functions (DGA2007), held in Olomouc, Czech Republic.

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It is evident then that two conditions are necessary to fix a and further that in the cases above given these two conditions are sufficient: we shall see later that two conditions are straight line, not always sufficient. When one condition the line) Thus have if is we (e. g. a point on the line, or the direction of given, a relation between the constants can be found. >) lies on the straight line, we Aa + Bb + C = 0. THE EQUATION OF THE FIRST DEGREE 38 one of the independent terms of the other, and the equation of the line can be found in a form which involves only one unknown or It is then possible to find the value of ratios or constants in undetermined constant.

And we obtain M (+p)-2|Liy+jp-i. = the undetermined constant appears in the second degree. (A'x + Vy+V) = represents, for different values of ^ Q (i) a pencil of lines passing through and of the the point of intersection straight lines f / If we give jm increasingly large values, the A'x + y + C 0. Ax+Hy+C = = S straight line represented by (i) approaches the position of the straight It is sometimes said that for an infinite line A'x+By + C'= 0. It coincides with this straight line. correct, however, to write equation (i) in the form value of /JL it \(Ax + By+C) + n(A'x + B'y+C') is = safer and more Q.

T; , on 1 = ax -} (ii) meets tho by'+l. 23. If the sides of a parallelogram arc the straight lines u equation sides of a triangle arc of any point P(jt'\ 1 O (ae, (# cos #)/-{ (ysin0)/& is b'\ meet the straight (3, 2) to lines are bisected: lines w= 0, through the Find the equation of the straight line which joins the point of section of u = 0, v = 0, to that of u -f ic = 0, r + ww = 0. 24. inter- four sides of a quadrilateral have equations ui v = 0, r-i w> = 0, -u = 0, t;~ w = 0. Find the equations of its diagonals.

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