434 51b EARNSHAW, on THE DIFFRACTION OF AN 



which taken between the proper limits becomes 



h\ ( Sir, ^ „ aeacota »+ocota 



<^.r. Jcos — (vt — B — — ~ \-' — ^ .0") 



a^TT I X b b ' 



Stt, , „ ScoCOta p — OCOtn A 



— cos~{vt — B + — ~, + ^ . x)i ; 



and the integral of this, taken with regard to x between the limits before 

 mentioned, is 



h\' 



s,n -{vt-B + -J-)- sm - [vt -B- -L ^ ) 



■i^TT- I P + (/ cot a 



sm — [vf — B + ~- - sni — -{vf — B ^ j-^ 



\ V 1 XV ft 



p — g cot a 



Let us now refer the image on the screen to polar co-ordinates, which 

 will be done by writing r cos 6, r sin Q for p, q respectively. For brevity, 



also, write V for -^ [vt - B + ~\, and 2m for , f^ .— , or its equal 

 XV b I ftsmaX ^ 



., ; then the above expression for the disturbance at M may be written 



a' sin g f sin V- sin { V- 2m sin (a + 9)} sin V- sin { F- 2m sin (a - g) j l 

 4»M'sin0'\ sin(a + 0) sin (o - 6) J 



By expanding the numerators of these fractions, and arranging the 

 result in two terms containing respectively sin V and cos F, this ex- 

 pression for the disturbance at 31 may be written in the following 

 form : 



a' sin a fl - cos {2m sin (a + 6)] _ 1 - cosj2ot sin (a - e)\ \ . 

 4 m' sin ' \ sin (a + 0) sin (a - d) ] ^^" 



^ ffl' sin g | sin|2>» sin (g + 6)} _ sin {2m sin (g - 0)}\ ^^ 

 4»«* sin 9 [ sin (g + 0) sin(a-0) j ^"^^ 



