Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

LIGHT PASSES THROUGH A LENS. 307
PROP. 20. Every thing remaining as in the last problem,
except that, close to the hole, a lens is placed of such focal
length that light diverging frorn A will be made to converge
to O: to find the intensity of light on the screen.
80. From (44) it appears that the form of the front
of the wave after refraction by the lens will be a sphere of
which O is the center. Let 0, fig. 22, be the origin of
co-ordinates: p and q the co-ordinates of a point M on the
screen: x, y,, those of P, z being parallel to 0A. Then
PM2 (p - )2 + (q -y)2 + z.
But by the equation to the surface of a sphere,
x + y2 + z2 = b".
hence
PMl2 = b2 + p2 + q2 2p - - 2qy,
P + 2 px qy
and PM = b + neary.
2b b b
The terms depending on tV2 and y' will be insensible, as
they will be multiplied by the very small quantities p2 and q2.
fp2 + q 2
Put B for b + ---: then, as in (73), the whole displace2b
ment at M is
27r/ ( r px qy
f~fvsin ' t - B + b + y.
This expression is much simpler than that of (75), as there
are no terms involving,2 and y:. The first integration
can always be performed: it gives
bX 2rf p qy\
-— cos (vt — + b
27rq b b
and if y' and y" are the least and greatest values of y for
a given value of x (given by the equation to the aperture
in terms of x), the first integral is, between these limits,
b> f 27rf pv qy' 29 p0 qy — il
-- cos5I- B) s- cos t-B i- +
27irq\ X b b ) X \b b
20-2