Dr Searle, Experiments with a plane diffraction grating 97 



plane of the grating. Then cos NOJ = 0, and the difference be- 

 tween the tangents vanishes, i.e. the two curves touch at J . 



Since the distance of ZLX from T is constant, the curve 

 corresponding to the vertical cross-wire is a small circle passing 

 through J with T for pole. At J the small circle is perpendicular 

 to the great circle TJ and the value of tan 1^ can be verified by 

 spherical trigonometry. 



The horizontal cross-wire is represented by the great circle 

 XT, but now both e^ and t] vary and no simple construction is 

 available for the v:liole curve through J corresponding to this wire. 

 The curve touches at ./ the small circle passing through J and X 

 with L as pole, and cuts at right angles the great circle LJ . Hence 

 tan III = cot LJZ, and then (26) can be verified by spherical 

 trigonometry. If we find cos LJZ and sin LJZ and divide, we 

 obtain the alternative form 



cos 6 — sintff sin 6 



tan iu = . : : 7P • 



cos i/f sm o) sm v 



If the angle between the two small circles which intersect in J 

 is A, then A is the supplement of LJT. But LJ = ^tt — 6, 

 JT = eg, LT = \tt, and hence 



cos A = tan 6 cot eg. 

 In the experiment cos eg has the constant value =F iXjd, and thus 

 cos A depends only upon d. The angle A will vanish when 

 cos A = 1, and this occurs when d = eg, i.e. when 6 has its critical 

 value. 



We can make visible a finite arc of the small circle w4th L for 

 pole. If we illuminate with white light the small opening across 

 which the wires are stretched, the position of J on this small circle 

 will be different for the different colours. The short length of cross- 

 wire will correspond for any colour to a small arc of a curve 

 touching the small circle at practically its middle point. The 

 envelope of these small arcs will be the small circle itself. The 

 image of the horizontal wire will thus be a dark curved line running 

 across the spectrum. 



§ 11. The critical values. The critical position of the grating 

 is reached when 6 = eg, and we have, by (20), (22), the critical values 



cos e, 



sm ijjc = sm eg cos eg, sm oj^ = i , 



(1 — sin^ egcos^eg)^ 



cos e., ,^ J , cos^ eg 



tan a;. = — 9 , (tan i v)c == —■ " • 



^ sm^ eg sm eg 



For the grating used in § 12 and with * = 1, 



cos eg = 0-33568, sin eg = 0-94198, eg - ^tt - 19° 36' 49". 



Then ^1 - ]8°26', co^ - 20°43' 18" and (7^.)^ = 6° 49' 17". 



VOL. XX. PART I. 7 



