85 
of y and 0 , and contributing when r = x to none of the co- 
efficients Yo, U_i, &c., when n is even. 
Also as 
2 -^ 
^ p dr ^ dd‘^ 
all the other terms in will contribute terms to Vo, U_i, 
&c,, which vanish upon integration with regard to B from 
0 to 27T. 
As all the other terms can be put into the above form it 
follows that, in so far as terms in the integral arise from the 
lower limit of r, the resultant displacement is in any case 
that in a plane polarised wave, and that is so however close 
the point considered may be to the wave front which is 
broken up. 
lY. 
We have now to consider the influence upon the integral 
displacement of the terms arising from the boundary. There 
are two cases : first, when the distribution of the sources is 
continuous around their origin, over a portion of the plane 
whose boundary is given by a continuous curve for which 
r = R, either a constant or some function of 0, and secondly, 
when the distribution is over a sectional part of such portion 
of the plane bounded by radii through the origin. That is, 
upon the terms from the upper limit in the first, and from 
the lower limit in the second of the integrals. 
Considering B. constant we shall obtain from the first of 
these integrals terms such as 
= »(»-!){ 1 - 5 } 
