356 Maj. Barnard on the Dynamic Theory of the Tides. 
w= (uy) — uy cotan 0—~7 5 6) 
(The negative sign is used since w represents the elevation posi: 
tive or negative, “above the undisturbed surface, ani 
increase with 4 and a, in the preceding discussion, there will be 
a fall of tide.) 
Referring back now to equation (8), Mr. Airy has shown (par. 
85, 86, ‘Tides and Waves”) that each term multiplied by S may 
be put under the general form 
© cos (t¢-++-ka) F 
in which 9 is a function of @ alone: and also that “the equation 
between w, u, and v; those between p’”, wand v; and that be- 
tween w, ", ‘and the terms arising from the disturbing foree, 
being all linear, we may take the terms arising from the istur 
ing force separately, and, finding the solution for each term, we 
may add all together. It will be sufficient, therefore, to proceed 
with the solution of the equation” (instead of equation (8)) 
0=9 cos (tt-+ka)—gw +p" 
and combining this with equations (4), @) and (6), 
— 6b? we +2n b2 sind pata 
dp _ 
ae = 
dp du d2 
a, ae 2 ee neg Hy bp Sue 
> hes 2nb sin 8 cos 6 >> 52 sin qe 
dv 
w= — 5 (uy)= uy cot0—7 5s 
we have Laplace’s differential equations of tidal motions (as given 
by Mr. Airy). my 
ae general solution of these equations is scarcely to be 
for; itis a matter of difficulty to find, ina very ne 
particular integral which will satisfy them. Beppe” ‘oe ms | 
here ”) And the particular Adis essayed by Lap. | 
Mr. Airy is of the TOES form | 
=a cos (vt = ko) 
u ree cos ((t+ka) 
v =c sin(tt+hka) 
“ae a cos (vt He 
of motion &c. of the tides eg from the set sane orces 
pressed by the particular term. Vues 
