87 
whatever function of 0 R may be. This requires that no 
term independent of x shall enter into any of the functions 
r_i, r-i. 
This condition is radically different from that proposed 
by Professor Stokes (On the Dynamical Theory of Diffrac- 
tion, Mathematical and Physical Papers, vol. II., page 288). 
Under the integral sign we should get 
and Professor Stokes’ argument is that provided that no 
finite portion of the boundary is a circular arc the function 
sin27rX~^ (at-W) will change sign an infinite number of 
times, and that having a mean value which is zero, the limit 
of the integral will be ultimately zero. That this is so, 
appears clear, and yet I think that we must stipulate also 
that it shall be ultimately zero for any portion of a circular 
arc also, as since R appears only in the form R/X in the 
trigonometrical terms, the limit bears no inverse ratio to the 
distance of the circular arc. And if we can imagine a por- 
tion of the arc small compared with a wave length, its 
effect at a finite distance from the origin would be finite 
compared with the length of the arc. 
VI. 
In the next place, the only terms in u_i which contribute 
towards Vq, are those terms in which y and 0 do not suplicitly 
appear. These terms therefore in and and 
those which must appear with them in order to satisfy the 
equation 
^ ^ — -0 
dx^ dy'^ dz 
may be looked upon in the same way as a particular integral 
in solving a differential equation, and the other terms which 
may appear may be considered as a complementary function, 
Neglecting these latter terms at present we get 
