ON TIDES. 303 
Then 
F = ku," | cos at | cos ot 
‘es 18 ku {cos ot , cos 3ct cos 5ot 
== hug see = SOR oe " 
Tw 3 1.3.5 3.9.7 
eae cos (2n-+ l)ot | 
a Or (2n — \eetl@ata orf r 3) 
The term involving cos 3ct corresponds to the remark of Ferrel. 
Each term in the frictional force will, of course, generate a tidal constituent of 
equal speed, so that the relations (2) will be disturbed, but such disturbance will 
usually be relatively small. 
Now all the constituents indicated by (3) have usually been allowed for in the 
practical harmonic analysis of observations, though often under the heading of 
shallow water constituents. But we next proceed to indicate the existence of con- 
stituents which do not appear to have been considered hitherto. 
Suppose that 
} u—=uycosct-+ujcosot ,8—=0 , . = - (4) 
while 
ot =710) ot =m oa) ke : ; : (5) 
n being an integer; then 
F=£ | up cos nO + wu’, cos (n+1)0| {up cos nO + w’o cos (n + 1)0} 
and this is periodic in 0, so that 
F =4a, + a cos0+ 6, sn @+....-+4a,cos'0+46, sin" 0+.... . (6) 
‘It is a straightforward matter to elaborate formule for the coefficients of (6), but we 
will restrict ourselves to the approximate form where w,’/u, is small. We have 
UW = (Uy" + Quy’ cos 9 + w9'2)8 cos (nO+) , 
where 
noe 
u 
tane = 0 us. 6 
Uy + wy’ cos 0 
and if we neglect ¢ we obtain 
F = k(u,? + 2uyu’) cos 9 + w’,?) | cos nO | cos nO 
cos nO , cos 3n8 
oe gage 
cos (27 + 1)n0 
= 5 bu? + u0") { 
i eet act uid] l 
et)" Or 1) eee 
8k, { cos (n — 1)0 + cos (n + 1)0 
= a Ug 9 | <= 3 pe ae 
fs cos (3n — 1)0 + cos (3n +1)0 _ 
; . 1.3.5 y ir caae 
The constitueat of (7) of speed o has the coefficient 
Sk a 
8x (uo? + U7), 
result which is of value in calculating, for example, the frictional force of M, speed 
ue to a combination of M, and §, currents. 
But some of the constituents of (7), e.g. that involving cos (n—1)0, may have 
speeds other than those present in the astronomical disturbing forces or in the 
1923 ¥ 
