w The Point.

Supposing 02 to be successively perpendicular to OY, OX, and

TT*P crgt

* sin 0 = a?' sin (0 - a) + y' sin ((9 - a'), (45)
y sin 0 = %' sin a -f yf sin a'. (46)

If both systems are rectangular, we have

»-5> *'-£ + «,

and the equations are

# = #' cos a - ?/' sin a, «/ = x' sin a 4- y1 cos a,

which are the same as equations (43).

EXERCISES.

I . If we transform from oblique co-ordinates to rectangular, retaining
the old axis of a? ; prove Y— y sin w, JST= # + y cos «.

2. If a-, y » 2*9 y' ^e the co-ordinate of a point referred respectively to
rectangular and oblique axes having a common origin ; prove that if the
axes of the first system, bisect the angles "between those of the second,

x = (x' + y'} cos|w,
y = (#'-?/') sin J&>.

3. Show that both transformations are included in the formulae —

y = \'x 4- n'y + v,
by giving suitable values to the constants A, p, &c.

*4. If the old axes be inclined at an angle «, and the new at an angle w',
and if the quantic asp -f Zhxy f i^2, referred to the old axes, be transformed
to aS? -f 2h'X7+ VY~, referred to the new ; prove—

(47)

a + & - 1h cos oj a' + 5' — 2A' cos w'
2 . ___