88 
which satisfy and therefore rj^ ^ 
will satisfy 
dr] 
dx^ dy^ dz 
Also from the values j^Yq and ^Yq we get 
and the condition that we may have no backward wave is 
that if we change the sign of a; in / and then put r-x 
these functions shall vanish. Hence the conditions are 
A(-i) = o/,(-i) = o. 
The terms which we have now found are those which 
must essentially enter in order to give the requisite value 
of the displacement. Before considering the rotations 
of the displacement, we will consider what may he called 
the complementary function, which will contain arbitrary 
constants, and which will generally only contribute 
terms to the integral displacement which depend upon the 
nature and shape and the finite distance of the boundaries. 
The terms which they contribute would generally be in- 
sensible, since in them we should have under the integral 
sign such expressions as sin27r\-^ (at — 'K), where since Bis 
to be measured in terms of the wave length, in all practical 
cases the mean value will be zero. 
I have already shewn that in the case of diffraction at a 
sharp angle, infinite terms would be introduced, and it will 
be noticed that this is essentially so with regard to It 
will be noticed that $ is perpendicular to the wave front, 
and we must either consider that a finite solution of the 
problem fails in such a case, or shew that treated as the limit 
