240     ON THE  COMPOSITION AND  RESOLUTION   OF STREAMS  OF
so == S (flj cos ttj sin (</> + €_1 - ft) - 6_1 sin ^ cos (<£ + e_a - ft)} y = $ {o_t sin cq sin (<£ + e^ - ft) + 6_t cos a_t cos (<£ -}- e_1 - pj},
where 8 denotes the sum of the expression written down and that formed from it by replacing the suffix 1 by 2. To form the expression for the intensity, or rather what would be the intensity if the quantities c and e were absolutely constant, not merely constant for a great number of successive undulations, we must develope the expressions for x and y so as to contain the sine and cosine of $ + k, and take the sum of the squares of the coefficients. k is here a constant quantity which may be chosen at pleasure, and which it will be convenient to take equal to e_1 — ft. If / be the intensity, in the sense above explained, or as it may be called the temporary intensity, we find, putting S for e,-ft__€l + ft, J = {a_1 cos a_x + a_2 cos a_2 cos 8 4- 1_2 sin a_2 sin S}^2
4- {— &_t sin tfj 4- a_2 cos a_2 sin S — &_2 sin a_2 cos S}^2 + (a_x sin a_x -f a_2 sin a_2 cos S — 6_2 cos a_2 sin S}^2
4- {6j cos o^ + a_2 sin a_2 sin S + &_2 cos a_2 cos S}^2
- a/ + V + < + V + 2 (a_A + 6_46_9) cos (a_2 - aj cos S
+ 2 (a^ 4- a_a&_t) sin («_2 - a_t) sin S.
On putting for a, 6 their values c cos /3, c sin /3, this expression becomes
^1 = ^ci^2 + ^ca^2 + ^2cA (^cos (^a_2 - ^ai) ^cos (/3_2 - /3i) ^cos ^s
+ sin (a_2 - otj) sin (/3_2 + /3_X) sin 8} ... (9).
In order that / may be independent of the difference of phase p2"~ Pi> ^an(^ therefore of S, we must have either
cos (a, ~ a_x) = 0,           sin (y3_2 + &) = 0,......(10),
or sin (a_2 - a,) = 0,           cos (p_9 - /3_L) = 0,......(11).
The equations (10) give «_2- a,= ± 90°, £_2 = - £,, so that the ellipses described in the case of the two streams are similar, their major axes perpendicular to each other, and the direction of revolution in the one stream contrary to that in the other. It will be easily seen that the equations (11) differ from (10) only in this, that what are regarded as the first and second principal planes of the second stream when equations (10) are satisfied, are accounted respectively the second and first when (11) are satisfied.