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```1915]         STATIONARY,  TO  THE  THIRD  ORDER  OF  APPROXIMATION             313
and a like equation in which a and a' are interchanged.    In the terms of the third order, we take
a. = P cos nt + Q sin nt,        a? = Pf cosnt + Q' sin nt,.........(45)
so that
«2 + a'2 = \$ (P2 + Q2 + P'2 + Q'2) 4- i (P2 + P'2 - Q2 - Q'2) cos 2nt
+ (PQ + P'Q') sin 2nt. The third order terms in (44) are
% (P- + P'2 + Q2 + Q'2) (P cos nt + Q sin nt)
+ 2 cos nt cos 2n* |i P (P2 + P'2 - Q2 - Q'2) - ^ (PQ + P'Q')}
(                                             OT2P              •                                     }
+ 2 sin nisin 2nt UQ (PQ + P'Q') -        (P2 + P'2 - Q2- Q'4
\                                              *J                                                     )
+ 2 sin nt cos 2nt J£Q (P2 + P'2 - Q2 - Q'2) + — (PQ 4- P'Q')} \                                          J                      j
+ 2 cos nt sin 2nt j^P (PQ + P'Q') + ^ (P2 + P'2 - Q2 - Q'2)} , \                             J                               )
of which the part in sin nt has the coefficient
Q (i (P2 + P'2) +1 (Q2 + Q'2)) + iP (PQ + P'Q')
+ n2/<7.{Q(P2 + P'2 -Q2- Q'2)-2P(PQ + P'Q')}
or, since n" = g approximately,
Q {f (P2 + P'2) - HQ2 + Q'2)} - f P (PQ + P'QO..........(46)
In like manner the coefficient of cos nt is
P{|(Q2 + Q/2)-i(P2 + P'2)}-fQ(PQ + Pm.........(47)
differing merely by the interchange of P and Q,
But when these values are employed in (44), it is not, in general, possible, with constant values of P, Q, P', Q', to annul the terms in sin nt, cos nt    We
obtain from the first
op
n« = g +1 (P2 + P'2) - HQ2 + Q'2) - |o (PQ + P'Q'X ......(48)
and from the second
^ = ^ + |(Q2+Q"0-i(P2 + P'2)-||(PQ + P/Q/);......(49)
and these are inconsistent, unless
(PP'+QQ')(PQ'-P'Q) = 0......................(50)
The latter condition is unaltered by interchange of dashed and undashed letters, and thus it serves equally for the equation in a'.os if + £ sin t',    a' = A' cos if. + J3' sin t', with                &' = -aa',    &=£(a'2-a2),    c' = 0,    c = 0.
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