MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. 
523 
have 
and consequently 
f ^^cdda— r ^cfda, (‘- 1 -) 
C. 
^.jjdda 
0 , 
sin — a-(n^ cos a^n-^ ’ 
and from the condition ^y=k§. 2 , 
k=^ r^^a^da 
0.\Qay 0 
^iga^ 
1 — ajWi cot fljMj 
( 22 .) 
9. Let now an indefinitely thin canal of fluid be conceived to reach from the 
centre in a straight line to the surface of the nucleus, and consequently to pass 
through the surface stratum having the thickness da^. Conceive this stratum to con- 
sist of an infinite number of elementary parallelopipeds, having their bases resting 
against the shell’s inner surface. As the oscillations of the shell’s surface are small, 
the simultaneous contraction of all these parallelopipeds can take place only in a 
direction perpendicular to their bases, and consequently da^ will become k^da^^ k^ being 
the cubical contraction of a mass of the fluid composing the stratum. The increase 
in volume of the shell, by the addition of the new stratum, will be less than the de- 
crease in volume of the nucleus by its abstraction from its mass ; and hence the 
nucleus must tend to expand in order to fill the empty space which would otherwise 
exist. This would evidently result as a necessary consequence of the cause of the 
variation in density of the strata from its centre to its surface. 
The radius of the nucleus after the solidification of its superficial stratum being 
a^ — kda^, and it being manifest that the entire mass of the nucleus before the solidifi- 
cation took place is equal to the mass of the solid stratum with the thickness kda^, 
and the mass of the new nucleus. 
/ ^a‘^da= f ^'a^da-\- j ^.jcdda, 
0 ^0 a^-kda^ 
or 
/ {^^—i)^^da— (^-/)aVa=0 
^ ax—kda^^ ^ 0 
where g>' is the density of the stratum with the radius a, after the solidification of the 
superficial stratum. As §2 and g are constant in the interval k^da, we have 
{^ 2 —§i)h(Ada ,= / {g—g’)a^da (23.) 
n 
But g being a function of a, and being what this function becomes after the ex- 
pansion of the stratum to which it refers, we may write 
f =/(«)» §'=Aa->^cAda), 
