242 



Dr. C. Chree. 



In obtaining this solution we have neglected terms in u 2 , i.e., terms 

 in (d£jdt) 2 or g 2 t 2 {p - p') 2 j{p + J/ ') 2 ' m tne expression (15), while there 

 appear in the solution terms containing g 2 t 2 (p - p')/(2p + p). Thus 

 our work is consistent only when (p - p')/p is small, and even when 

 this is the case the fact that u 2 increases as t 2 involves a restriction 

 which should not be overlooked. It would not, I think, be a very 

 •difficult matter to obtain a complete solution answering to the full 

 value (15) of p. Treating u 2 at first as a constant, we could at once 

 write down, from my general solution" 55 " for the isotropic elastic sphere, 

 the displacements answering to the surface pressure J/) « 2 (3P 2 - 1) ; 

 but the explicit determination of the corresponding supplementary 

 solution would be much more laborious than in the first case treated 

 above. 



§ 5. When p and p are equal, and u 2 is thus really constant, the 

 complete values of the stresses and displacements answering to the 

 surface pressure (15) are as follows : — 



xx = -II- gp\z + at) + \pu 2 + 



3 p'u 2 a 2 



[(7 + 2iy)a> 



yy 



zz 



n 



7 + 5>; 



+ 3r ] (x 2 + by 2 ) -3(7 + 6^ 2 ], 

 gpiz + ut) + ipu 2 + f ^g- 2 [(7 + 2 V ) a 2 



+ ■3^(5^ + ^-3(7 + 6^^, ^ 



n - g P '(z + ut) + \ P n 2 - » £2L*L [2(7 + 2 V )a 2 



-3(7 + v )(x 2 + f) + 6 1 z 2 l 



xy = -V^V^- 2 -[2(7 + 5r/)], 

 xz{xz = yzjyz = V«V~ 2 + |>(7 + 57;)] 



a/x = /3/y - 



(21); 



[n - ipu 2 + g P '(z + ut)] 



l-2>; 



Y (22). 



[(n - ipu 2 ) z + <^>fc +i(z 2 -x 2 - </)}] 



§ 6. In real liquids viscosity is more or less present, and as the 

 hydrodynamical equations have been solved for the case of an ellipsoid 



* ' Camb. Phil. Soc. Tram.,' vol. 14, p. 250. 



