SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 17 
The time t measures the rate at which the stress falls off, and is called (I believe by 
Professor Maxwell) “ the modulus of the time of relaxation of rigidity; ” it is the 
time in which the initial stress has been reduced to e~ l or ’3679 of its initial value. I 
do not suppose, however, that any solid conforms exactly to this law; but I conceive 
that it is often useful in physical problems to discuss mathematically an ideal case, 
which presents a sufficiently marked likeness to the reality, where we are unable to 
determine exactly what that reality may be. 
Mr. J. G. Butcher has found the equations of motion of such an ideal sub¬ 
stance from the consideration that the elasticity of groups of molecules is continually 
breaking down, and that the groups rearrange themselves afterwards.* These con¬ 
siderations lead him to the following results for the stresses across rectangular planes 
at any point in the interior, vix. (with the notation of § 1):— 
and similar expressions for Q, It, T, U ; where m —- is the coefficient of dilatation, n 
O 
that of rigidity, S the dilatation, and a, (3, y, the components of flow. 
These expressions are clearly in accordance with the above definition of elastico- 
. dS S (d/3 ,dy\ 
viscosity, for * + -,=»( 
If the expressions for P, S, &c., be substituted in the equations of equilibrium 
of the elementary parallelopiped, it is found by aid of the equation of continuity 
<p__<Za f 7, that when inertia is neglected 
dt dx 1 dy 1 dz’ & 
1 d\-l 
t ^ dt 
m 
l V ■ d ' 
-H+ m * 
f+”V 2 « 
4I=o 
and two similar equations. 
By the same reasoning as in § 1, we may put, S= ; —, and the equations become 
1 m d dp 
t'm — \ndt_ dx 
+x=o. 
Then supposing the substance to be incompressible, so that m is infinitely large com¬ 
pared to n, and therefore mA-m—is unity, the equations become 
dp (I d' 
-i 
V 3 a + X=0 
and two similar equations. 
* Proc. Lond. Math. Soc., Dec. 14, 1876, p. 107-9. It seems to me that the hypothesis ought to repre¬ 
sent the elastico-viscosity of ice very closely. 
MDCCCLXXIX. 
D 
