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.

FRINGES SHIFTED BY INTERPOSING A GLASS. 285
of the pencils of light: to find the alteration produced in
the fringes of interference.
59. Let T be the thickness of the glass: and consider
the case of Prop. 12. It is plain that, as in (42), the
length of that portion of the path of one pencil which traverses the glass is not to be estimated by its linear measure,
but by,u x that measure, inasmuch as the motion of the
wave is slower in glass than in air by that proportion.
We must consider therefore that instead of describing the
space T in air, the wave describes a space equivalent to gA T
in air; and therefore the effect of the glass is the same
as that of lengthening the path by (g - 1) T. Instead of
HM in the expression of (47) we must put HMl+(u - 1)T:
and the intensity of light at M is now
LO.cos GM - IM - ( - ) T,
which as in (48) is changed to
4c2 2r (asin a -1 T). cos- --- OiM -
(a + b) ' \ a+b 2
and the places of maximum brightness are now determined
by making
asin -i X
nOLa01- T =, or=, or= X, &c.:
a + b 2 2
or by making
_ -tb a+b
OM^ a b. (-1) T, or a.{( 1) T:},
2asina 1)2a sin a
a+b
or = a - 1) T 2X}, &c.
2a sina
60. Now if, - 1 were the same for rays of all colours,
it is evident that these expressions would be precisely the
same as those found for the bright points in (48), increased
by a constant
a + b 1.
2 a sin a