92 Dr Searle, Experiments with a plane diffraction grating 



?2 sin 6 + 712 cos 6 = cos 17 = l-^ sin 6 + n-^co&d. ...(12) 



We also have, by (7), if P^OT = e^, P^OT = e^, 



cos 62 = ^2 = m^ T i^^jd = cos e^ T iA/(? (13) 



Hence m^ is known at once. Using ^2^ = 1 — '^^^ — ^2^, we have, 

 by (12), 



sin^ ^ (1 — m^ — ^2^) = (cos r) — n^ cos ^)2. 



Solving for n^ and taking the negative sign in the ambiguity, we 

 have 



sin i/f = ^2 = cos 6 cos 7^ — sin ^ (sin^?] — m.^^)^. ..-(14) 



Using this value of Wg in (12), we find 



^2 = sin ^ cos ?7 + cos ^ (sin^ 17 — ^2^)* (15) 



Since cos NOP^ = I2 cos 6 — n^ sin 6, we find from (14) and (15) that 



cos iYOPa = (sin2 77 - mg^)* (16) 



Thus the negative sign has been correctly chosen in (14) for the 

 transmitted beam, since for this cos NOP2 must be positive. If the 

 positive sign is used in (14), cos NOP2 is negative, corresponding 

 to the reflected diffracted beam. 



In terms of i/j and co, the direction cosines of OPo are cos i/j cos to, 

 cos i/j sin 60, sin i/j. Hence Wg = cos j/f sin co, and thus 



sin to = mg/cos i/j = cos e2/cos i/f (17) 



The two angles ifs and co completely determine the direction of the 

 diffracted ray. 



In the experiment of §§ 8, 9 the incident rays are parallel to 

 the axis OX of Fig. 3. Hence ?j = 1, m^ = 0, n^ = 0. We then have 



cos 62 = W2 = T iX/d, (18) 



cos Tj = sin 9, (19) 



and thus, since sin^ 7] — cos^ e^ = sin^ eg — cos^ 77, 



sin ijj = n^ = sin 6 cos 6 — sin 6 (sin^ ^2 — sin^ 9Y, . . .(20) 



^2 = sin^ ^ + cos ^ (sin2 62 - sin2 ^)* (21) 



sin 60 = ma/cos ifj = cos eg/cos ijj (22) 



Since e^ and 9 may be restricted to be less than hr, we see that 

 no diffracted beam will be formed if 9 exceeds its critical value e^. 



§ 8. Apparatus. The general arrangement is shown in Fig. 4, 

 The grating G is attached to a horizontal shaft A, with its plane 

 parallel to A and its ruhngs perpendicular to ^. A horizontal 

 colhmator L has horizontal and vertical cross-wires intersecting 

 at C in its focal plane ; these are illuminated by a sodium flame S. 

 The straight Hue joining C to the appropriate nodal point of the 



