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11)11]                                    UYmtODYNAMlOAL NOTES                                           13
the order a".    Tlui tcrniH to be retaine.d in (2), in addition to those given in (5), are
2/3 ( 1  2//) cos 2&' 4* 47 cos U,r 4- 4a^ COH ,7;
=; 2/8 cos 2.r  2a/Q (cos ,7: 4- UOH IJ,i;) 4- 47 cos & 4- 4<a/3 COH no. Expressing the terms in cos x by moans of//, we. got finally If* - 'Itjij  C.OIIHL. 4- 2/y ( 1 - // - a"  ex4 + ft)
+ (o4 4- 2/9) COH 2,7; 4- (47 - :| a"1 - 2a) CON Ik- ....... (<))
In order to satisfy the surface condition of constant pressure, wo must
take
0--A'4.        7=tl-X'    ........................ (10)
and in addition
1 _. ,, _ aa - j| *    0 ............................ (II)
correct to oa inclunive.    The expression (I) for -^ thus annumes l,hi^ form
^     i/ - ctr " ci >H ,-r + ^ a% """ CM IM 2./-  (1Q aV "/' c.os ,'k; .......... (12)
from which // may be calculated in terms of,/; as Jar an aa inclunivc.
By HuecesHive approximation, determining ^ HO us to make the mean value of i/ equal  to tfero, we find HH far w a4
// "(a + ^aa)coH,r -(| a51 * |a4) COM 2,r { j| ;t COH .'i.r - j^ COM 4r,    ..,(1H) or, if we write an before a for the coellicient of COM,/',
//    rf coH.r -~ (|<fa+ ||f/4);oH2: 4- PscoH!U' - (^/4coH4a;,     ...(14) in agreement with equation (20) of Htoken' Supplement, KxpreKMed in teniiK off/, (II) boeonioM
t/ I -- |f/* .............. , ....... . ........ (16)
or on reMtonitiun uf A*, <*,
g-kP-fatW- \fatW.    .................. .,....(10)
ThuH tin* extension of (8) in
r>f;/Ml 4-A%+|^rt4) .......... . .............. (17)
which aim i agrees with HtokcH* Supplement,
If we purMue the approximation one Hinge further, we find from (12) terms in of, additional to thtwu c!XprKHtjtl in (Hi).    Tluwe are
873             243                125

It i of intentst to cotnpire thts potential and kinetic energies of waves
* [I9lfi.    Brnide (Prat;. /,</. if nth. .Voe. Vol. xv. p. J0, 1010) throws doubts upon the utility of Stulten* series. )
Ilished in 1880J the same author treated the problem by another method in which the space coordinates x, y are regarded as functions of $, i/r the velocity and stream functions, and carried the approximation a stage further.