372 THE THEORY OF THE HELMHOLT2 RESONATOR [401 Our special purpose is concerned with the difference in the values of ^ on the two sides of the surface r = c, and thus only with the difference of 2's. We have 2 , 6 , 1 -tan£<9 2 (inside) - !> (outside) = ^ - log cos ^ + log IT^- • 3 9 , /-, • o\ 3 « - log (1+am 2]--^--5 v 175 • v \ -T -i 7— ...(38) In the application we have to deal only with small values of 9 and we shall omit lc-tf, so that we take 2-39., it will indeed appear later that we do not need even the term in 6, since it is of the order &2c2. In pursuance of our plan we have now to assume a form for U over the circular aperture and examine how far it leads to agreement in the values of ^ on the inside and on the outside. For this purpose we avail ourselves of information derived from the first approxi-A niation. If C, fig. 1, be the centre and CA the angular radius of the spherical segment constituting the aperture, P any other point on it, we assume that U at P is proportional to {CA2 - CP2}~*, and we require to examine the consequences at another arbitrary point 0. Kg. 1. Writing CA = a, CO = b, PO = 6, POA = (j>, we have from the spherical triangle cos CP •=• cos b cos 6 + sin b sin d cos (p, or when we neglect higher powers than the cube of the small angles, ......................... (40) </>)2, . . .(41) Thus CA-- CP2 = a2 -ba--62- 2bd cos <£ = a? - b2 sin2 <f>-(6 and we wish to make sin 0 d6 d^ [2 (in) - 2 (out)] A as far as possible for all values of b, the integration covering the whole area of aperture. We may write 6 for sin 6*, since we are content to neglect terms * [Except as regards the product of sin 0 and the first term on the right of (39), since the term in 02 is in point of fact retained in the calculation. W. F. S.]ces to make