188 
MR, Ct. H. DARWIN ON THE STRESSES 
I. 
THE MATHEMATICAL INVESTIGATION. 
§ 1. On the state of internal stress of a strained elastic sphere. 
Let there be a homogeneous elastic sphere, for which co —yu is the modulus of com¬ 
pressibility (or incompressibility, as I shall call it) and v the rigidity.'" Take the 
centre of the sphere as the origin for a set of rectangular axes x, y, z. Let the sphere 
be subjected to no surface stresses, let it be devoid of gravitation, but subject to 
internal force such that the force acting on a unit volume of the elastic solid is expres¬ 
sible by a gravitation potential IF;, a solid spherical harmonic of the i tix degree of the 
coordinates x, y, z. 
Let iv be the density of the elastic solid, a the radius of the sphere, and r the radius 
vector of any point measured from the centre of the sphere. 
Sir William Thomson has investigated the state of internal strain produced under 
the conditions above described. If a, (3 , y be the displacements his solution is as 
follows :— 
where 
a= (EiO? - Fit*) 'F - G<r*+*4 ( 1 ) 
' ' Clfj ' 
(hr 
E, 
i[(i + 2)co —o] 
■2(i-X)v{[2(i +l) 2 +1 ]a> - (2 i + 1) v} 
Tji (i +1)(2% + 3)© — (2% + l)u 
2^i + l)v{[2(-i + l) 2 +l>-(2i+l>} 
~ ico 
(2 i + l)v{ [2 (i +1) 2 + l]a) - (2z + l)v} 
> 
and similar expressions for j3 and y.t 
Now let P, Q, It, S, T, U be the six stresses, across three planes mutually at right 
angles at the point x, y, z, estimated as is usual in works on the theory of elasticity. 
Let P, Q, Pi/ be tractions and not pressures, and let be the hydrostatic pressure at 
the point x, y, z. 
Then P+Q+B being an invariant of the stress quadric, we have, 
P = — i(L+Q+P) 
if S-zzi—f —j- —I— . so that S is the dilatation, then according to the usual formulas,! 
ax ay dz 
* The pliraseology adopted by Thomson and Tait (first edition) and others seems a little unfortunate. 
Oue might be inclined to suppose that compressibility and rigidity were things of the same nature; but 
rigidity and the reciprocal of compressibility are of the same kind. If one may give exact meanings to old 
words of somewhat general meanings, then one may pair together compressibility and “pliancy,” and call 
the moduli for the two sorts of elasticity the “ incompressibility ” and rigidity. 
f Thomson and Tait’s 1 Nat. Phil.,’ § 834, (8) and (9) ; or Phil. Trans., 1863, p. 573. 
+ Thomson and Tait’s ‘Nat. Phil.,’ § 693. 
