the Slit in Interference Experiments. 473 



plane bisecting the acute angle 2o> between the mirrors ; 

 then the equations of the planes of the mirrors may be 

 written 



a? sin (0 — gj)+2C0S (0 — co) = 0, 



x sin (0 + co) + z cos (6 + a) = 0. 



Suppose the slit perpendicular to the plane through the 

 centre of the slit and the line of intersection of the mirrors, 

 it edges being perpendicular to the plane of xz, and let a be 

 the distance of the central line of the slit from the axis of y. 

 The coordinates of a point of the slit distant f from this 

 central line are 



x = asin 2#-|-£cos 20, y , z = a cos 20 — £sin 20, 



and the coordinates of its image in the first mirror are 



x — 2 sin (0 — co){x sm (0 — (o) -f-^ cos (0 — co)} =a sin 2w + £cos 2ft), 



Vo, 



z — 2 cos (0— G>){# sin (0 — co) +.c cos (0 — &>)} = — a cos 2&) + | sin 2<y. 



Hence, the propagational speed of light being taken as unity, 

 the undulatory time of passage, from the point x , y , z to 

 the point x, y, b, of the stream reflected at this mirror is 



y i= ={(# — asm2co — £cos 2&>) 2 + {y — yo) 2 + {fi-\ a cos 2co — £ sin 2ft)) 2 } 2 



= ■{(6 4- a cos 2o)) 2 — 2 (,r cos 2&) + Z>sin 2&))£ + (# — a sin 2<o) 2 



+ (2/-i/o) 2 + f}» 



a? cos 2» + 6 sin 2ft) -. 1 (.i 1 — a sin 2ft>) 2 + (?/ — Vn) 2 + ? 



==6 -f a cos *2ft) — — t—. 5 £ + ~=r r^ ^ — L_ ^ — - — 



Hfl cos Zft) z 6 + a cos 2ft> 



The undulatory time of passag'3 V 3 between the same two 

 points of the stream reflected at the second mirror is 

 obtained from V! by changing the sign of g) ; hence the relative 

 retardation, measured in length in air, is 



A = V,-Y 1 = 2 Bi f" (ig+ , fflJ;) - 

 o f a cos 2g) 



Fresnel's Biprism. — Let the plane through the edge of 

 the prism perpendicular to the flat face be the plane of yz, 

 the edge being parallel to the axis of y, and suppose that the 

 flat face is turned towards the slit, which is placed in the 

 plane £ = with its central line along the axis of y. 



If a be the distance of the edge of the prism from the 

 origin, and a lf a 2 be the acute angles of the prism, the 

 equations to its inclined faces will be 



z = a — tan a^a?, 



~ = a + tan « 3 a?. 



