﻿478 Prof. D. N. Mallik on the 



31. From these we may easily derive the corresponding 

 rotational equations, viz.: 



mto'x + fiil'u =nvV, • • • • (12) 



where w x , H z are the curls of u, v, w, U, V, W. Boxissinesq's 



theory is thus seen to he capable of being interpreted as 

 being based on the postulate of twists, defining the disturbed 

 state of the medium. 

 For, if we write 



we get, putting 



m 

 a 



/+A=c 2 A 2 /, 



provided A 2 (n*-A) = (13) 



l7r o^ Itt d# 



Further, the equation of equilibrium of a material medium 

 regarded as an elastic body would be tj 2 Q.j: = 0, &c, so that 

 (13) amounts to the statement that in forming the equation 

 of motion we must regard the material medium to be at 

 rest. 



Again, U and u are assumed by Bonssinesq to be connected 

 by an equation of the form U=/(w), and in particular, for 

 dispersion, U is taken by Boussinesq 



=\m+Cv; -\-D\/ 2 k, where \, 0, D are constants; (11) 



whence on our notation, 



n.e = \co x + l)v 2 co x , &c, .... (15) 

 or 



A =/ + D/ -+ 2 A where 2f= »*(1 +X) + nJ~ - 1 Y (46) 



If we admit that / is an harmonic function, the equation 

 can obviously be written in the form 



A+p =a f, (17) 



(he constants beiDg suitably adjusted. 



