250 
MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 
_ 
u = 
V = 
^ = 
Ifi 
a dfjL 
0 f JiY ]' 
00 lv/(l-/^^)K 
1 Byjr 
2o)afj, v/ (1 — fx-) 00 ’ 
(1 — ^ 
2a)afx dfx ’ 
v' —gC 
(34). 
where, as there is no longer any ambiguity, we have omitted the bars from the 
symbols U, V, 0, v'. 
Since 0^/0^ = 0 when the motion is steady, we see that the first equation is 
identically satisfied if we suppose h, 0 both independent of 0. In this case we may 
take 
C=tC,'P4ix) .(35), 
where the constants C,i are arbitrary, and deduce 
whence 
0 = — S gnC,F,, (p); 
■vr _ _ \/ (I A''^) ^ p 
^ ~ 2ma^ <?/<■ 
(36). 
The last equation gives the velocity which must be imposed on the particles of 
water in latitude sin”^ju, in order that the free surface may be maintained in the form 
defined by (35) without any external force. We see that it is theoretically possible 
to maintain an arbitrary surface-form by correctly distributing the longitudinal 
velocities of the fluid particles. If however the series (35) involves harmonics of 
odd order the value of V given by (36) becomes infinite at the equator, and to 
prevent a flow of liquid across the equator it would be necessary to impose an 
infinite velocity on the particles of water there. Hence, if the water extend either 
wholly or partially over both hemispheres, the distribution of velocity and the form 
of free surface must be symmetrical wuth respect to the equator, at least so far as 
concerns that part of the ocean which communicates across the equator. 
Conversely, any arbitrary initial distribution of longitudinal velocity symmetrical 
with respect to the e(|uator may be rendered permanent by an appropriate adjust¬ 
ment of the free surface. These results hold good whatsoever be the law of depth, 
provided it be a function of the latitude alone. 
If 0 be not supposed independent of 0, we see from the second and third of 
equations (34) that the velocity of How across any element ds inclined at an angle y 
to the meridian is 
