80 PROCEEDINGS OF THE AMERICAN ACADEMY. 



For solid spheres rising in the magma we may compute the " critical 

 radius " (/{), that is, the radius of the largest sphere which would obey 

 the law of the Stokes formula. The values for I?, as stated in the fifth 

 column, have been found with the help of Allen's formula : ^^ 



9 v^ 



2 gd (d - d')' 



The terminal velocities of the solid spheres having the critical radii 

 would be, for the corresponding values of R, v, and (d — d'), as follows : 



The figures show that even for small solid spheres the velocities are 

 considerable. With increase of radius the terminal velocities would at 

 first increase very fast, and then more slowly. However, since the 

 resistance to the motion would, for large spheres, vary with the square 

 of the velocity, neither the Stokes formula nor any other yet developed 

 can declare the actual velocity for large solid spheres moving in the 

 magma. 



Nevertheless, Allen's formula for large spheres is of distinct help in 

 guiding one to a proper appreciation of the case. It reads : 



,,„ 1 4 7r d — d 



^ =k-s-^"-d-^ 



where ^ is a constant for a given liquid-solid system. ^^ Jt follows that 

 the terminal velocity here varies directly as the square root of the 

 radius and as the square root of the difference of the two densities. 

 Referring to the table showing terminal velocities for solid spheres with 

 critical radii, it seems clear that, in any of the three cases, spheres of 



M H. S. Allen, Phi). Mag., 60, 324 (1900). 

 33 H. S. Allen, Phil. Mag., 50, 532 (1900). 



