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.

INTERFERENCE OF UNDULATIONS. 249
and the quotient of the latter by the former gives
a sin A + b sin B
tan C =
a cos A + b cos B
Tle form of the expression
csin -(vt- ) + C}
shews that the length of a wave is the same as in either
of the undulations compounded, but the difference of value
of C from A and B shews that the maximum of vibration
for a given particle does not generally take place with the
same value of t as for either of the undulations compounded. The magnitude of the maximum vibration, which is c or
V/\a2+ b2 + 2ab cos ( - B)
depends on the value of A-B: its greatest value is a +b,
when A - B = 0, and its least value is a - b, when
A - B = 1800. In these two cases C is equal to one at
least of the two angles A and B.
Pnop. 6. To examine the effects of interference of two
equal and similar undulations: and to shew that wlen one
is (p +- x length of the wave behind the other (p being
a whole number), they will destroy each other.
15. In the case of equal vibration, a= b. The value
of c is then
~2 2,A-B
/{2a2 + 2a2 cos (A - B)} = 2acossin A + sin B A + B A + B
and tan C = = tan,- or Ccos A + cos B 2 2
If A-B =0, the value of c is 2a, and C= A: that is, if
we add two such undulations as (/3) and (') (fig. 2), we
shall have an undulation in which the maxima are at the
same places, and the maximum vibration is double what it
was before. With any other value of A - B, c is less than
2a: and when B = A: 180%0 c is 0, that is, there is no