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 - 2aŁ) 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- P«scoH!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!Xpr«KHtjtl in (Hi). Tluwe are
873 243 125
It i« of intentst to cotnpire thts potential and kinetic energies of waves
* [I9lfi. B«rn»ide (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.