312 Lord Kayleigh on Laplace's 
the law of force between any two elements is always the same, 
<p(r), as a function of the distance, and that the difference 
between one fluid and another shows itself only in the inten- 
sity of the action. The coefficient proper to each fluid may 
be called the " density," without meaning to imply that it has 
any relation to inertia or weight. The force between two 
elements (of unit volume) of fluids I. and II. may thus be 
denoted by p x p 2 <p(r)', that between two elements of the same 
fluid by /3j (f>(r), or p\ <]>(r), as the case may be. 
We will first examine the forces operative in a fluid whose 
density varies slowly, that is to say undergoes only a small 
change in distances of the order of the range of the forces, 
supposing, for simplicity, that the strata are surfaces of revolu- 
tion round the axis of z. The first step will be to form an 
expression for the force at any point on the axis. 
The direction of this force is evidently along z y and its 
magnitude depends upon the variation of density in the neigh- 
bourhood of O. If the density were constant, there would be 
no force. We may write 
. dp ^(Pp z> #p x\ 
or in polar coordinates, 
a dp Q d 2 p r 2 cos 2 cPp r 2 sin 2 6 . , ... 
8p=£rco S e+-^ 2 — J — + ^ 2 — r - + termsmr 3 .(4) 
For the attraction of the shell of radius r and thickness dr 
we have 
277-r 2 dr <p(r) f "fy cos 0sm0d0=~ r 3 <p(r) dr&+...; 
and for the complete attraction, 
-Q- -j 1 ^ 3 <K r ) dr + terms in 1 r 6 <£(r) dr. 
6 dz Jo Jo 
The difference of pressure corresponding to a displacement dz 
is found by multiplying this by p dz. Thus 
dp= T d £) r*<p(r)dr+... 
and 
Pi—p2= "a" (p*—pf) I T% 4>(. r ) dr + terms in I r 5 cp(r)dr. (5) 
