The elements of coordinate geometry, by S. L. Loney, M.A.

CHANGE OF AXES.

113

*132. To change from one set of axes, inclined at an
angle o, to another set, inclined at an angle o', the origin
remaining unaltered.
Y YY
p
N _ _ _
0 M N L X
Let OX and O Y be the original axes, OX' and OY' the
new axes, and let the angle XOX' be 0.
Take any point P in the plane of the paper.
Draw PN and PN' parallel to 0 Y and OY' to meet OX
and OX' respectively in N and N', PL perpendicular to OX,
and N'M and IV'M' perpendicular to OL and LP.
Now
z.PL = z YOX = o, and PN'M'= Y'OX o'+ 0.
Hence if
OV= x, NP=y, ON'=x', and N'P=y,
we have y sin o = NP sin Lo = LP = MN'+ 1Jf'P
= ON' sin 0 + N'P sin (w' + 0),
so that y sin o = x' sin 0 + y' sin (o'+ 0)...............(1).
Also
x + y cos = ON+ NL = OL = OM + N''
= ' cos 0 + y'cos (w' + 0)...............(2).
Multiplying (2) by sin o, (1) by cos o, and subtracting,
we have
x sin o = x'sin ( - 0) + y' sin (w - w'-0)...... (3).
[This equation (3) may also be obtained by drawing a perpendicular from P upon OY and proceeding as for equation (1).]
The equations (1) and (3) give the proper substitutions
for the change of axes in the general case.
As in Art. 130 the equations (1) and (2) may be obtained by
equating the projections of OP and of ON' and N'P on OX and a
straight line perpendicular to OX,

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