438 Mh EARNSHAW, on THE DIFFRACTION OF AN 



Hence, 

 „/sin'>«A sinse , sin30 , sin 30 , SlsinSe + 4sin^3e 

 ^(-/ ) = -4--'" -T:6--"'^4.-8:1()-^"- 4.5.6.7.8.9.10'" ^ - 



Wherefore, by substitution, we finally obtain 

 y_3a' m^ m' (81 + 4 sin" 3 0)^'' 



16* "6""^8.10 5.6.7.8.9.10 ^ '"^ 



-^ oi ^Uli l'''Y «'V (rV (81 + 4sin'3g)aV' /rv" 

 ~16"^'^ eT^'V / ■*■ 8.10.Z»'U/ 5.6.7.8.9.10.6" [\l "'"•"f 



A striking feature of this series is, that its leading terms are entirely 

 independent of 9, and therefore while r is so small as to allow the series 

 to be represented by its first three terms, the brightness will be independent 

 of 6 : and therefore consecutive circles of uniform brightness will suiTound 



the centre. 



3 

 When r is =0, the brightness = —^ a', which is independent of X, 



and therefore the central point of the image is white. 



When r is so large as to require the fourth term of the series to be 

 taken notice of, the circles which correspond to those radii will have 

 their brightness diminished by a term of the form sin^30: they will 

 therefore be most bright when sin 39 = 0, that is, where they are inter- 

 sected by the rays drawn upon the screen as before mentioned ; at points 

 more remote from those rays the brightness will gradually diminish, and 

 be least when sin 39 = 1, that is, at those points which lie exactly between 

 them. 



2. Let us now examine the image in the neighbourhood of the six 

 rays ; for this purpose 9 must be supposed small, and Z must be expressed 

 in a series ascending by powers of 6. Upon this hypothesis we find 



Z = J—; \m" - -y= . sin (;« -x/3) + - sin'' ( ^ ) > + terms involving 0-, 9'... 



For any of the six rays we may write 9 = 0, and therefore the bright- 

 ness of any one ray is accurately expressed by 



