1917] AND ON THE THEORY OF FOUCAULT's TEST 467 the alternatives following the sign of 6 (f), with exclusion of the case <f> = 0. If 0 is finite, 2 Si (<££) may be equated to TT, and we get / = 47r2 (1 or 0), according as 9 <£ is positive or negative. But if <£ = 0 absolutely, Si ($£) disappears, however great f may be ; and when $ is small, I = 47r2 cos" p '+ 4 sin2 p {2 Si (<££)}2, in which the value of the second term is uncertain, unless indeed sin p = 0. It would seem that the difficulty depends upon the assumed discontinuity of R when 9 = 0. If the limits for 6 be + a (up to the present written as ħ 6), what we have to consider is + 00 r r+a. sn - in which hitherto we have taken first the integration with respect to 0. We propose now to take first the integration with respect to £, introducing the factor eħ^ to ensure convergency. We get 2 sin (T - R) cos (0 + c/>) . . .(33) There remains the integration with respect to 0, of which R is supposed to be a continuous function. As /x tends to vanish, the only values of 0 which contribute are confined more and more to the neighbourhood of <p, so that ultimately we may suppose 0 to have this value in R. And u,d6 tan" tan"1 which is TT, if ^> lies between + a, and 0 if </> lies outside these limits, when /A is made vanishing small. The intensity in any direction 0 is thus independent of R altogether. This procedure would fail if R were discontinuous for any values of 0. Resuming the suppositions of equation (31), let us now further suppose that the aperture extends from £ to £2, where both |x and £2 are positive and & > fi- Our expression for the vibration in direction 0 becomes sin T [cos -p {Si (0 + </>) % + Si (0 - 0) f } + sin p {2 Ci (0£) - Ci (0 + <£) | - Ci (0 - </>) £}]g + cos T[cos p {Ci (0 - £) f - Ci (61 + 0) £} ......... (34) We will apply this to the case already considered where £20 = 500, £0 = TT ; and since we are now concerned mainly with what occurs in the neighbourhood of ^ = 0, we may confine <j> to lie between the limits 0 and £0. Under these circumstances, and putting minor rapid fluctuations out of account, we may 302the maximum illumination near the edges is some 6 times that at the centre.n unity. Under the same conditions the Ci's in (21) may be omitted, so that