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To find the length of the perpendicular from a given point upon a given straight line.

Instead of equation (2), in article 48, we must use the equation just found, and then proceeding as usual we shall find

(y-ax-b) sin. w

P = ± √{1+2 a cos. w+a"}"

It will be concluded from an observation of these formulas, that oblique axes are to be avoided as much as possible; they may be used with advantage where points and lines, but not angles, are the subjects of discussion. As an instance, we shall take the following theorem.

52. If, upon the sides of a triangle as diagonals, parallelograms be described, having their sides parallel to two given lines, the other diagonals of the parallelograms will intersect each other in the same point.

Let A B C be the triangle, A X, A Y the given lines, EBDC, CFAG,

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HAIB the parallelograms, the opposite diagonals DE, FG, and HI

will meet in one point O.

Let A be the origin, AX, A Y the oblique axes

x1y, the co-ordinates of B

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Y2a (x2-x) at E

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Equating the values of y in (1) and (2) we find X=

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also equating the values of y in (2) and (3) we find the same value for X; hence the abscissa for intersection being the same for any two of the lines, they must all three intersect in the same point.

Similarly we may prove that if from the angles of a plane triangle straight lines be drawn to the bisections of the opposite sides, they will meet in one point.

CHAPTER IV.

THE TRANSFORMATION OF CO-ORDINATES.

53. Before we proceed to the discussion of equations of higher orders, it is necessary to investigate a method for changing the position of the coordinate axes.

The object is to place the axes in such a manner that the equation to a given curve may appear in its most simple form, and conversely by the introduction of indeterminate constants into an equation to reduce the number of terms, so that the form and properties of the corresponding locus may be most easily detected.

An alteration of this nature cannot in the least change the form of the curve, but only the algebraical manner of representing it; thus the general equation to the straight line y = ax + b becomes y=ax when the origin is on the line itself. Also on examining articles 46 and 51 we see that the simplicity of an equation depends very much on the angle between the axes.

Hence in many cases not only the position of the origin but also the direction of the axes may be altered with advantage. The method of performing these operations is called the transformation of co-ordinates.

54. To transform an equation referred to an origin A, to an equation referred to another origin A', the axes in the latter case being parallel to those in the former.

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substituting these values for y and x in the equation to the curve, we have the transformed equation between Y and X referred to the origin A'.

55. To transform the equation referred to oblique axes, to an equation referred to other oblique axes having the same origin.

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Let the angle x Ay=w, xAX= 0, xAY=0'; Draw N R parallel to PM, and NQ parallel to A M, then y = MPMQ+Q P=NR+QP

sin. NAR

sin. PN Q

AN

+ PN

sin. NRA

sin. PQN

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59. These forms have been deduced from the first, but each of them may be found by a separate process. The first and last pairs are the most useful. Perhaps they may be best remembered if expressed in the following manner.

Both systems oblique, the formulas (55) become

y={X sin. XAx + Y sin. Y Ax}

x = {X sin. X Ay + Y sin. YA y}

1

sin. x Ay

1

sin. x Ay

Both systems rectangular, the formulas (58) become

y= X cos. X Ay + Y cos. Y Ay
x= X cos. X Ax+ Y cos. YA x.

If the situation of the origin be changed as well as the direction of the axes, we have only to add the quantities a and b to the values of x and y respectively; however, in such a case, it is most convenient to perform each transformation separately.

If the new axis of X falls below the original axis of x, the angle must be considered as negative, therefore its sine will be negative and its cosine positive. Hence the formulas of transformation will require a slight alteration before applied to this particular case.

Since the values of x and y are in all cases expressed by equations of the first order, the degree of an equation is never changed by the transformation of co-ordinates.

60. Hitherto we have determined the situation of a point in a plane by its distance from two axes, but there is also another method of much use. Let S be a fixed point, and S B a fixed straight line; then the point P is also evidently determined if we know the length S P and the angle PS B.

If S Pr and PSB = 0, r and 0 are called the polar co-ordinates of P. S is called the pole, and SP the radius vector, because a curve may be supposed to be described by the extremity of the line S P revolving round S, the length of S P being variable. The fixed straight line SB is also called the axis,

To transform an equation between co-ordinates x and y into another between polar co-ordinates r and e,

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Draw SD parallel to A X, and let the angle B S D=9, and the angle YAX=w.

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and if the origin A be the pole, we have a 0 and b = 0.

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Of these formulas (3) and (4) are the most useful.

62. Conversely, to find r and ✪ in terms of x and y: from (1) we have

sin.{w—(0+)}

M

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sin. w cot. (9+)~cos. w ;

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66

where the symbol tan.-1a is equivalent to the words a whose radius is unity, and tangent a."

circular arc

also r2 = (x — a)2 + (y — b)2 + 2 (x − a) (y — b) cos. w...

.(30)

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