387 



midway between two rigid boundaries may break in two, one half migrating each 



way (1 ). Such features can only be handled by a more complete analysis In 



which the shape of the globe is not limited by assumption but Is left to the 

 control of the hydrodynamlc action. 



DESCRIPTION OF THE MATHEMATICAL METHOD FOR DETERMINING MIGRATION 



Provided the gas globe remains spherical, the effect of gravity 

 alone is easily found. The analysis can be extended to Include the effect of 

 plane boundaries by using the method of images that is familiar in electro- 

 static theory, and by expanding in negative powers of the distances from the 

 surfaces, as in Herring's report (U) The entire analysis, including a spe- 

 cial method of approximation for the first oscillation, has been completed 

 (2) but is not given here. 



If the solution Is required to satisfy the boundary conditions only 

 as far as the Inverse second powers of the distances from the boundaries, dif- 

 ferentia] equations of the following type are obtained: 



+ J R^R (n^X + NyY + N^z) - I [g^iX - X,) + gy{Y - Y,) + g,{Z - zA 

 X = \n,R'R - I I^ f^R^R'dt - ^[R'dt [2] 



Y = ^NyR'R - ^^ f\*RUt -^ /rUi [3] 



Z = ^N.R'R - ^^f\*R'dt - ^[R'dt [U] 



Here R is the radius of the bubble, 



X, Y, Z are the cartesian coordinates of its center, 

 t is the time, 



R, X, y, Z stand for dR/dt, dX/dt, dY/dt, dZ/dt, respectively, 

 Xq, Yq, Zq denote values at ( = 0, 



p Is the density of the water, 

 Pq is the total hydrostatic pressure Including atmospheric 



pressure, at the point X^, y^, Z^, 

 Pg is the pressure of the gas In the globe, supposed to be a 



known function of iJ, 



