238     ON THE  COMPOSITION AND  EESOLUTION  OF STREAMS OF
>-(*).
then our four equations become
cos ft cos 7j. ff1 -f sin ft sin yl. \ -f cos /32 cos 72. g2
+ sin /32 sin 72. 1\ =    cos /3. #,
cos ft cos 7j. Aj — sin ft sin % • & -f- cos ft cos 72. Aa
— sin ft sin 72. #2 =    cosfi. h,
cos ft sin ry1.g1- sin ft cos % . A, + cos yS2 sin 72. #2
— sin ft cos 72. \ = — sin /?. A,
cos ft sin 7i. Aj + sin ft cos fy1.ffl + cos ft sin 72. A2
+ sin /32 cos 72. ^2 =    sin /3. ^.
Multiplying the first and second of these equations by 1, V(—1), and adding, then multiplying the third and fourth by — V(~l)j 1> and adding, and putting generally
(5),
-.(6),
we have
(cos ft cos 7j - J - 1 sin /31 sin 7,) Grl + (cos /32 cos 72
- y - 1 sin ft sin 72) (72 = cos /3 . £, (sin ft cos 7j — x/ — 1 cos ft sin ^j) ^ + (sin ft cos y2
- >/ -1 cos ft sin 72) (72 = sin y8 . # ; which two equations are equivalent to the four (4).
Putting for shortness plt p2, ql, q2 for the coefficients in the left-hand members of equations (6), we have
G
-        ' " v ;'
q2 cos    - p2 sn        p, sn    - q, cos /      p^ -On substituting for pv q2, &c. their values, we find
Ms - PA = cos (7, ~ 72) sin (ft - ft)
+ J~l sin (7l - 7,) cos (ft + ft).
Now the equations (6) cannot be incompatible or identical unless the above quantity vanish.    But this can only take place when
sin (/32 ~ ft) = 0 and sin (^ - 7,) = 0, or else
cos (ft + ft) = 0 and cos (% - y2) = 0, or lastly
sin (ft - ft) = 0 and cos (ft + ft) = 0.