By Luther Pfahler Eisenhart

This textbook explores the configurations of issues, strains, and planes in house outlined geometrically, interprets them into algebraic shape utilizing the coordinates of a consultant element of the locus, and derives the equations of the conic sections. The Dover variation is an unabridged republication of the paintings initially released by means of Ginn and corporate in 1939.

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**Example text**

If we say that direction numbers of any segment of a line are direction numbers of the line, it follows that a line has an endless number of sets of direction numbers, but that the numbers of any set are proportional to those of any other set. 2) by x\ and y\, being of the first degree in x an equation of a line through the point (x\, y\} and with direction numbers u and v. 1] // y 2 -yt = v. 1). Since parallel lines , ; by definition, it follows that y2 ) are direction (xi 9 y\) and y 2 any line - xi = u, and (x 2 , any two points on a y\ are direction numbers of the line, parallel to line, and x2 x\ also of it.

3) ju 2 1 < X < + 1 ; when and, conversely, any two numbers X, /x satisare direction cosines of a line. conditions these fying X 1 ; In Figs. 4) = cos A, fj, cos B. FIG. 8 FIG. 5) The equations x = xi + d\ y = yi + dp are parametric equations of the line through the point 28 (x\, y\) Angle between Sec. 6] and with X and direction cosines distance from and is Two Lines /z; the point (%i, y\) to the parameter d is the a representative point positive or negative according as the latter above or below P\ along the line, or to the right point or left of PI when the line is parallel to the x-axis.

That the coordinates of any point on a line through the numbers of the line. origin are direction Show 8. 5) hold for a line parallel to either if we make the convention that "the angle between" coordinate axis the positive directions of two parallel lines 9. Show zero. is that for the lines in Figs. 3) A + cos 2 A = l. is 2 equivalent to the trigonometric identity sin 10. Show that (x ~ * i)( * 2 - *3) + (y ~ y^* - *) = o two lines through the point (xi, y\) parallel and perpendicular respectively to the line through the points (# 2 yz) and are equations of , (*s, y*)- Given the 11.