294 The Rev. Professor O'Brien on the Laws 



V — ^ Jnt-k{px+sz)} V^^^ 



2 ' 



V _ ^~^^' ^ ^{nt-kCpx-sz)} V^ 

 ' s + jiAs' 2 ' 



V'= S ^-l^^ I A -^^{n^-*'(/^+»'r)}A/~. 

 Xs + fxs' J 2 

 12. We shall now proceed to deduce real solutions from 

 these imaginary integrals, and first in the case of vibrations 

 perpendicular to the plane of incidence. 

 Assuming 



= -Pe-' 



we find 



s—tx^s' 



Vi -p^-^'v/^y 



Itt^s'^ 





k^s^ 



Hence 



( \^Ve-»^-J \ 



Vi-p^-^^-iy 





We have, therefore, for y'lhrations perpendicular to the plane 

 of incidence, 



y _ _ ■p^-4V^f_g{nt-k{px-sz)}VZri^ 

 where P, 5, cr and x ^^'^ given by the two equations 



We may now immediately find another set of values of V, V, 

 and V, by simply changing the sign of the \/ — 1 in these 

 formulae, which gives us 



