SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 25 
bhk { 1 + f-\ ms OU- 
But this same slice, in its undisturbed condition, had a volume bhk. Therefore the 
equation of continuity is 
cos [m(x'-vt) + e]. 
Now the hydrodynamical equation of motion is approximately 
dp _ dV d~p 
dx' dx' dt? 
The difference of the pressures on the two sides of the slice PQqp at any depth is 
cLt) 
NwXy- 7 ; and this only depends on the difference of the depressions of the wave- 
surface below the axis of x on the two sides of the slice, viz. at P and p. Thus 
dp dp 
dx' ^ dx' 
d 2 P 
Substituting then for rj from the equation of continuity, and observing that — - 7 is 
• CvJbCLtXs 
d 2 P 
very nearly the same as —, we have as the equation of wave motion, 
7 d 2 P , ^ . r / / x n dV , d?p 
Cjh dP +VUJ E Sm ~ Vt ) ~dP + dP 
But 
So that 
dV 
dx 
— = — TO F sin [to (x — vt) -f- e] + TO G cos [to (x — vt)-\- e]. 
C ^=gh ( -jp-\-m{G cos[to(x / — vt)-\-e\— (F —Ey) sin \m(x — vt)-\-i\}. 
In obtaining the integral of this equation, we may omit the terms which are 
independent of G, F, E, because they only indicate free waves, which may be 
supposed not to exist. 
The approximation will also be sufficiently close, if x be written for x r on the right 
hand side. 
Assume, then, that 
' £= A cos [m(x —^) + e]+B sin [to( a?— vt)- be]. 
By substitution in the equation of motion we find 
— m 2 (v z —gh) {A cos+ B sin } = to { G cos — (F— ’Eg) sin). 
And as this must hold for all times and places, 
MDCCCLXXIX. 
E 
