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.

300 UNDULATORY THEORY OF OPTICS.
cerned in the integration, the whole displacement may be
expressed thus:. 2r r a+b ap aq 2
s in -(vt -B) fy cos-. - x - - +
\ X a6 w a+b ai + a + b\
- cos —(vt-B)fj;,sinr-. + b-
X\ \ ab [kk a + b Y a+b
Let the integrals be E and F: then the whole displacement
2 7r 27r
=E. sin -(v t - B) - F. cos- (V - B),
À À
and hence as in (17) and (23)} the illumination is represented by E2 + F2.
74. Now
f 7r a +b ap 2 C O + b aqa \2
E == fLy Lcos - * --- tî? - --- x cos -. --- x y - a - sy -. X ab ab a b
sinr a+b (f ap a + b ( aaq b }
- s-. - x- x sin-. - -
X ab a + b' X ab a +b
proceeding therefore according to the precepts of (73), the
first thing to be done is to find
_ cos y. - and f. sin < -
[fw a-t-b / aq } d.f7r a+b ( aq \'~},y ~ ab - a+b) and ab Y a+h)
These integrals cannot generally be found in terms of y.
Tables however have been formed* expressing the numerical
value of the integrals
' Suppose U to be the function of unknown form which is the integral of S
a given function of s. Then the values of U corresponding to s - h and s + h
dU d2U h2 d3U h3
are U —h+ —. - — +&c.
ds ds2 2 ds3 2.3
d U + d2U h2 d3U h3
and U + h+ ' + s3. + &c.
- ds ds2 ~2 ds3 ~2.3