Now if x and y be function of a third quantity 0

1

2

2

and d ̧2 y = (d'x) {do2 y dp x—do2 x dp 7 }

0

Now x = r cos 0 .. d x = r cos 0 -r sin and y = r

=
y de r sin xr cos 0 and d2 x =

d

2

r cos 0

Ꮎ

dr e-
r sin -r cos 0 and dy = d2r sin + do

0+ do r cos 0

r sin 0

r cos

Let x2

ax + b be the quadratic factor, where a 4 b, then the roots x2 ax + b = o are imaginary; also let V, = (x2- ax + b) Q where Q is an integral function of x, and does not

(x2

powers

ax+b) or separating in the first member the odd and even

of x.

ax + b) where R (x2) and R' Write ax b for x2, then the

R (x2) + x R' (x2) = P(x2 (x2) denote integral functions of x.2 second member vanishes, and the first member may be reduced to the same form as before, R' (x2) + x R, (x2) each function being diminished one half,—and A and B still entering only in the first degree. By repeated substitution of ax b for x2, we shall at last arrive at a result of the form M + Nx, which since it is satisfied by two values of x, furnishes two equations M = o and No for determining A and B; and then substituting the values of A and B, we have

- (Ax + B) Q

Cx + D x2+x+1

(1)

.. x + 2 = (Ax + B) (x2 + x + 1) + (Cx + D) (x2 + 3)

Make x2 = 3

+ C

2 x cos 0+1 be a quadratic factor in the deno

points in the image are similarly situated with respect to the mirror as those in the object, its form will be exactly similar to that of the object. Again the rays falling from the image at q, upon the mirror CD, will form the second image at q, equally distant with q, from the mirror CD, and of the same form. Hence the second image of Q will be formed at q1, at the same distance from the mirror CD, and inclined at the same angle to it as it is from the mirror AB, and as it is inclined to it. The first image will be inverted with respect to Q, but the second erect.

E

B

LM

Let AB be a plane mirror, and Q a given point and q its image determined by drawing QMq perpendicular upon the mirror and taking QM = qM. Join qB and qA and produce them to E and F respectively; then an eye must be situated within the < Bq A, so that it will be able to view the image at q.