12 Mr. Potter's Reply to Mr. Tovey. 



passing through the perpendicular line and this point. Then 

 rid ir will be an elementary area in the aperture, and for 

 simplicity I take the case of plane waves, the other only dif- 

 fering by a constant. 



Putting h and a the same as before, we have the displace- 

 ment of the particle at the point, arising from this element. 



rieir 



"" v'r- + /i^ + .r^ — 2 r x cos 6 



X sin-^ \vt — /v/;.2 + j^i jf. af — 2rx cos 6- \ 



Expanding and retaining the terms involving the first power 

 of X only, we have the whole displacement 



2 IT ax /^ p r^ cos 6 2 tt / , \ 



+ -IT J Jo WTh- "^^ —[vt-Vr-^ h^ 



p p r'^ cos 6 . 2Tr / , \ 



+ ^ ^' / A r sm — - ( u if — 'v/;.2 . ^2 \ 



Jrj6 (^2 ^ ^2^4^ XV ' f « y 



p r 6 . 2 TT / , \ 



= Vr-VWTl^ s.n_ (.^ - Vr- + /,2) 



2 TT a >r /» r^ sin ^ 2 tt / \ 



+ — T-y f^TT^ '°'ir (^^ - '^^"^^T^) 



y^r^sin^ . 2 7r/ \ 



+ axl -sin — Ivt — -i/,.2 j_ 12 1 4. c. 



and taken between the limits ^ = o and 6 = 2-n 



pi' . 2 77 / \ 



which is the same expression as on the line itself, and indi- 

 cates a maximum in the plane perpendicular to the line when 

 there is a maximum in the line. 



In examining the terms involving :i^, I shall retain only 

 the pruicipal term in the coefficient of that quantity, which 

 has K^ in its denominator, and thus the quantity which is the 

 correction of the last found expression is the following : 



2-17^ a x^ p P r^ cos^ ^ir f \ 



^^rytf(.2 + ^2+ ^s)|sin— (i-^ - v^rVAH..^) 



between the limits ^ = and ^ = 2 tt 



