WALLACE O. FENN 379 



cos A = = cos = 1 



n 



Since this subject has apparently never been worked out from the 

 point of view of surface energy it seemed desirable to be satisfied 

 that, when the forces of surface tension at the contact angle are in 

 equihbrium, the surface energy is also at a minimum. Unfortunately, 

 we have not been able, even with the expert assistance of a professional 

 mathematician, to express x and 5 of equation (1) in terms of the 

 degree of ingestion of a particle as measured by the length of the line y 



in Fig. 3, and thereby to obtain the value of — and finally an equation 



dy 



like (7) for a surface of any degree of curvature. Theoretically this 

 procedure is not impossible, but the necessary equations are too difficult 

 to solve, A test case has been taken, however, where the radius of the 

 particle, g, is one-quarter of the radius of the cell {r = 1) and the 

 values of Lx (the change in surface area of the cell in contact T\dth the 

 plasma), and of 5 (the surface area of G in contact with C), have been 

 calculated for the different values of y, the height of the spherical 

 segment inside the cell (Fig. 3).^ These values are given in Table I 

 together with the cosine of the corresponding angle of contact. By 

 assigning various values to m and n and using the calculated values 

 of ^x and 5 in the equation^ sm + A.tw = E, curves can be plotted 

 showing how the surface energy, E, varies as y increases to 0.5 or 

 twice the radius of the particle, G; i.e., as ingestion approaches com- 

 pletion. These curves, for different values of m when 11 = 1, are 

 shown in Fig. 4. The values of m used were calculated from the 

 equation for the contact angle equilibrium, m = — n cos A where A 

 is the angle of contact corresponding to the chosen value of y and n 



^ The complication arising from the increase in the radius of the cell to r + Ar, 

 as the particle is more and more completely ingested is responsible for the dilTiculty 

 of the calculation. In order to evaluate Ax and 5 it is necessary to solve for z 

 (Fig. 3) in an equation involving z' and z^ by trial and error. This must be done 

 to four significant figures in order to calculate A.v and s accurately enough for 

 the purpose. 



*This equation is the same as (1), replacing gTc-gTp by m and cTp by n as 

 in equation (6) and omitting the constant 4 wr-n = 12.566 n = (.v — A.v);; repre- 

 senting the surface energy of the cell when spherical. 



