490 FRANK F. GROUT 
as a larger mass than any sphere considered; and secondly, that 
the moving mass is not impeded by a liquid of uniform character, 
but on one side has a more viscous wall, while on the other side there 
is less resistance and some added tendency tomove. On the whole, 
however, such a calculation may give a fair idea of the order of 
magnitude of the motion. 
The rate of flow of liquids through a pipe may be compared 
with the rate estimated by this method. In comparison with the 
pipe, actual convection, though moving a larger volume of liquid, 
has the friction of only one solid wall. The evident error of 
estimating by settling spheres alone appears from the fact that a 
sphere of ro meters radius gives the same rate in viscous as in less 
viscous magma. On the other hand, the formula for flow in a 
pipe gives too much weight to the matter of viscosity. The best 
idea 1s probably obtained from a consideration of both calculations 
and a comparison with the observed rate of convection in lava lakes. 
Thermal convection.—On the basis of the data discussed above 
we may calculate the rate of settling of large spheres by reason 
of their greater density when cool. , 
Assumed temperature difference, too°C. 
Main magma specific gravity, 2.70 
Cool magma specific gravity, 2.71 
Density difference, .o1 
Final rate of motion of a sphere of to meters radius, nearly 1,000 meters 
per hour'. 
t The calculation for this rate of motion is given in detail for this case. Later 
estimates are made by the same method. Ifa small sphere sinks in a viscous liquid 
the final rate of motion is found by a formula of Stokes in Trans. of the Cambridge 
Phil. Soc., YX, No. 2 (1850), p. 8. 
Has 2gR? (d—d,) 
L= aGV. i 
where R is the radius of the sphere, d, the density of the liquid around it, g the accelera- 
tion of gravity, and V the viscosity. The largest sphere that will obey this law is 
calculated by a formula given by Allen in Phil. Mag., L (1900), 324: 
teins 
2gd(d—d,)° 
For larger spheres the velocities are proportional to the square roots of the radii 
sma R’ 
ere, 
In the case of thermal convection, the second formula becomes 
R= 9(5)? ne 
2(980)(2.70)(.01) 
