Dr Searle, Experiments with a plane diffraction grating 91 



Since rj^ tas the constant value r], while m^ or cos eg depends 

 upon A, it follows that, if white light is used, to each A there will 

 correspond a position of Pg on the small circle through P^ with L 

 for pole. 



§ 6. The deviation. If D is the angle (< n) between the forward 

 directions of the incident and the transmitted diffracted beams, 

 cos D = l-J.2 + ^1^2 + n-ffi^. If the plane ZOP^ (Fig. 1) cuts OXY 

 in OHi, where XOH^ ^ ^i, then, since P^OZ = rj, 



l^ = sin 7] cos ^1, m^ = sin 17 sin <^i, Wj == cos rj, 



and similarly for Pg- Thus 



1 — 2 sin^ ID = cos D = sin^ 77 cos (^^ — </>2) + cos^ 77, 



and hence sin |Z) = siniy sin | (^1 ^ ^2)' (1^) 



as can also be shown from the isosceles spherical triangle P^LP^ 

 in Fig. 2. 



In the case of the transmitted diffracted beam, l^^ is positive. 

 Noting that n^ = n^^ cos 77, putting m-^= a + h, m.^ = a — b, and 

 substituting for l^, l^, we find 

 2 (sin2 17) - 62) _ sin^T^ - a^ _ 52 _ ^,^^^,^2^ _ ^2 _ ^2)2 _ ia%^f. 



Thus sin^ ID is greater than b^ except when a = 0, and then the 

 two are equal. When a = 0, ni^ = — m^. If we take m^ positive, 

 we see, by (7), that, since i is positive, m^ = — m2 = iA/2(^. Hence 

 b = iXj2d. Thus, if Dq is the minimum deviation, 



sin iZ)o = iA/2«; (11) 



Since rj does not occur in (11), m^ = — m^ gives a minimum of D 

 for any given value of rj — a minimum having the same value for 

 all values of 77. 



§ 7. The sloped grating. For the experiment of §§ 8, 9 it is 

 convenient to use axes differing from those of § 4, Now let OY 

 (Fig. 3) coincide with OT and let OL make an angle 6 with OZ. 

 Then the direction cosines of OL are sin 6, 0, 

 cos 9, those of OT are 0, 1, 0, and those of ON 

 are cos d, 0, — sin 6. 



Let ^1, m^, ^1 and I2, m^, n^ be the direction 

 cosines of the forward directions OP^, OP^ of 

 the incident and the transmitted diffracted beams. 

 Let W2 = sin 0, so that the diffracted ray OP^ 

 makes an angle i/f with the plane OXY, counted 

 positive when P2OZ < In. Let the plane ZOP.^ 

 cut OXY in OH^ and let XOH^ = m. Then, if -q is the common 

 angle between OP^ or OP^ and OL, 



