96 Dr Searle, Experiments with a pJune diffraction grating 



If JK = ^, Pa^ = 'A'' ^^ = ^' -^'^^ = ^'' ^lien 



tanZ= limit of 75-^ = cosi/f f-y^ ) (23) 



If XH = a, PiH = y, the direction cosines of OP^, OP^ are 

 ?j = cos y cos «, m^ = cos y sin «, n-^ = sin y, 



?2 = cos if}' cos a>', m^ = cos j/(' sin to', ng = sin 0'. 

 Since ZOL = d, the direction cosines of OL are sin 6, 0, cos ^. But 

 P^OL = P2OL = 7], P-PT = ei, P^OT = 62, and thus the funda- 

 mental equations (13) and (12) become 



cos iji' sin oj' = cos 7 sin a =F iXjd, >. . . (24) 



cos ijj' cos ii)' sin d + sin ip' cos ^ = cos y cos a sin ^ + sin y cos 6. 



(25) 



The vertical coHimator wire corresponds to the great circle ZX, 

 and for this a = 0, but y varies. If ly is the inclination to JZ of 

 the corresponding path described by P^, 



tan/, = cos^(|^y(|^)^. 



Differentiating (24) with respect to y and then putting y = 0, so 

 that 0', co' become ifs, co, we have 



— sin i/f sin co {di/j'ldy)^ + cos ijj cos oj {dw'ldy)^ = 0. 

 Hence tan /r.= sin i/j tan co. 



The horizontal colhmator wire corresponds to the great circle"' 

 XT and for this y = 0, but a varies. Differentiating (25) with 

 regard to a and then putting a = 0, we have 



— sin iJj cos CO sin 6 {di/j'/da)Q — cos i/j sin co sin ^ {da}'/da)Q 



+ cos i/f cos ^ (difj' lda)Q =--= 0. 

 Hence, if Ih is the inchnation to JZ of the corresponding path 

 ofP,, 



tan /^ = cos 



cos xjj cos ^ — sin ifj cos co sin 6 



.(26) 



sm 0L> sm u 



Multiplying numerator and denominator by cos a>, and replacing 

 cos^ CO by 1 — sin^ to, we find 



T , T U cos d — Uo sin 6 .^„, 



tanZjj= tanZ^+ A ^ . „ 27) 



sm CO cos CO sm 



Since the direction cosines of ON are cos 6, 0, — sin 6, and I2, Wg 

 in (27) refer to OJ, l^ cos 6 — n^ sin d = cos iVOJ. In the critical 

 position, J lies on the great circle LT, which corresponds to the 



