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482                               ON  PERIODIC  IRROTATIONAL  WAVES                                [419
C + cos 4^ I

   f    37a7    25a"    15a27    ig s        ) + cos5a?|- -2() --^-f-^-^aS + Sej ................... (13)*
The constant part has no significance for our purpose, and the term in y can be made to vanish by a proper choice of g.
If we use only ; none of the cosines can be made to disappear, and the
value of g is
g^l-tf-M-Tof ............................ (14)
When -we include also /3, we can annul the term in cos 2# by making
' ........................... (15)
and with this value of $
    5a4    619a
But unless a is very small, regard to the term in cos 3# suggests a higher value of ft as the more favourable on the whole.
With the further aid of 7 we can annul the terms both in cos 2# and in cos %x.    The value of /3 is as before. _ That of 7 is given by
a"/      139a\
and with this is associated
The inclusion of S and e does not alter the value of g in this order of approximation, but it allows us to annul the terms in cos 4# and cos 5x. The appropriate values are
afi                 a7
480' and the accompanying value of 7 is given by
while jB remains as in (15).
We now proceed to consider how far these approximations are successful, for which purpose we must choose a value for a. Prof. Burnside took a = -}. With this value the second term of ft in (15) is nearly one-third of the first (Stokes5) term, and the second term of 7 in (20) is actually larger* than the
[* With the alterations specified in the footnotes on p. 481, the terms in (13) involving ct2/3?/, and (a7, a3/3) cos 3ar, become 2y .  a2/3, and cos 3z ( - ^ a7 + ~ a3/3 J.   Then the highest terms in
(16), (17), (18), and (20) become respectively - ^ a, ~ ( + ^ a? \ - ^ o, and ~ ( + ~ a the second term in (20) being now little more than half the first when a = J.    W. P. S.]*" + 2/3e~^ cos 2a?