246     ON THE  COMPOSITION AND RESOLUTION OF  STBEAMS  OF
'                        coefficients in some cases diminished in given ratios, and in some
cases suppressed altogether. Each stream has to be treated in a similar way. The final disturbance being resolved in any two
\                        rectangular directions, each component must be put under the
form 27 cos $ + Tsin <£, and the sum of the squares of U and V must be taken to form the expression for the temporary intensity. All the quantities such as U and V will evidently be linear functions of c cos e, c sin e, c' cos e', c' sin e', &c., where c, c'... and e, e'... refer to the different streams, so that U for instance will be of the form
Ac cos e 4- Bo sin e + A'c cos e' + B'c' sin e' + ...
where A, B, A', B'... are independent of the time.    The tem-11                          porary intensity will involve  172, but  the actual  intensity will
involve m(Z72), or [ j                              mS (Ac cos e -f Be sin e)2 + 2ttlS {(Ac cos e + Bo sin e)
[!                                                                                                     (J/c'cos e' + Fc'sine')}.
|                              Now   the   products   such   as   cos e cos e',   cos e sin e',   &c.   will
1                               have a mean value zero, since the changes in e and those in
J   ',j                             e' have no relation to each other, and therefore the expression
I   ;                               for m(f72) becomes
I    |                                         m2 (Ac cos e -h Be sin e)2, or 2m (Ac cose + Be sine)2,
I    i
that is, the sum of the quantities by which it would be expressed
' s '                             were the different streams taken separately.
;j                                        Two streams which come from  different  sources,  or which,
1 !( i                             though in strictness they come from the same source, are such
j !l j '                           that the changes of epoch  and  intensity in the one  have no
4 ",'                             relation to the changes of epoch and intensity in the other, may
I                            be called independent.
9.    Suppose that there are any number of independent polar-i                           ized streams mixing together; let the mixture be resolved in any
manner into two oppositely polarized streams, and let us examine the intensity of each.
Let us take one stream first. The intensities of its components are given by the formulae (13), which become somewhat simpler in the case of opposite polarizations, since $,= —&, and
I                          aa=90° + al.    Hence
j                                ct» = {sin2 (/8t + /3) sin2 (a, - a) + cos2 (ft - ft) cos2 (ax - a)} c2;