18 
LORD RAYLEIGH ON THE CIRCULATION 
we get from (76) in terms of real quantities 
u— coskx[— cos nt-\-e~^ + y^ [nt—Piy-^-yp}^ 
/V2 
sin kx 
Jh 
cos (nt—\ 7r)+e Wy+yi) cos {nt—^rr—P{y-\-yi)} 
(77). 
It will shorten the expressions with which we have to deal if we measure y from 
the wall (on the negative side) instead of as hitherto from the plane of symmetry, for 
which purpose we must write y for y-\-y v Thus 
u— cos&cc{ — cos nt-\-e p y cos (nt—fiy)} 
V= k ^/2 ^\ X ^' cos ( nt —i n )— 6 ~^ cos ( nt —i-v-Py) 
From (78) approximately 
V 3 i//=/3 v /2. cos kx e~ p y sin (nt — ^n—Py) . 
(78). 
(79), 
du , dv 7.7 
— -+•— = k sin kx cos nt 
dx dy 
(80), 
u 
d^± dy~yfr 
dx dy 
= ^k/3sh\2kxe ^(—cos Py-\-c ^)+ terms in 2nt . . (81), 
—^kfi sin 2kx e ^(sin fiy-\- cos /3y)-f-terms in 2nt . . (82). 
As in former problems the periodic terms in 2 nt will be omitted. For the non¬ 
periodic part of \p of the second order, we have from (66) 
k6 
vh//= — ^-sin 2kxe~ P!/ {sm /3y -\-3 cos Py — 2e~ Py } 
(83). 
In this we identify y 4 with —, so that 
, k sin 2 kx e~&, . _ , „ ~ . 
'/'=—j G --J— {e 1 n ftj +3 ''Of- fy + ' <• f, i 
• • • (84), 
to which must be added a complementary function, satisfying yh//=0, of the form 
siul1 (2 /j.— 2/)+ B (2/i—3/) cosl1 2 %i— 2 /)} ■ ■ ■ ( 85 )> 
