2 MR. AIRY'S SUPPLEMENT TO A PAPER "ON THE THEORETICAL 



and therefore the whole disturbance of ether on the point of the retina, produced 

 by that part of the pupil which is not covered by any retarding plate, is 



7^sin^(z;^-e + 4-.i/), 



the limits of the integral being the values of y corresponding to the boundaries of the 

 part of the pupil not covered by a retarding plate. 



But if a portion of the pupil be covered by a plate producing the retardation R 

 (expressed as an angle) in the phase of the wave, the expression to be integrated 

 through the limits proper for the covered part will be 



T^sin (^ (. / - f) - r) 

 or 



Let the limits of the pupil be from — A to + ^, without regard to the other ordi- 

 nate upon its surface (which amounts to supposing the form of the pupil to be a pa- 

 rallelogram), and let the part which depends on R be taken between the limits and 

 -|- h (which amounts to supposing that half of the pupil to be covered which is on the 

 side on which h is considered positive). Then the whole disturbance of the ether is 



J sin -^ ( V ^ — e -{-—y) from y = — htoy = 

 -\-J sin \-~ \vt — e-\- —yj — Rj from y = ioy = -\- h -^ 



\e C 21: ( bh\ 2% , ^ . , /27r, ^ , „\ 



-oos(?^(w-e + ^)-R)} 



The coefficient of cos -c- (v f — e) is 



2- 



Ke C 2'!t bh (2-n bh „\ l 



-;^{^««x~-i+^os^-cos(-^.--Rjj 



^e r 21: bh _, , 211 bh „ . • 2ir bh . „ ") 



— " 2Vb I A — cos ^ • y— cos R -1- COS -^ • y X COS R + sm ^ • — X sm R [► 



\e ]■/ 2ir bh\ . 2ir bh . „ ") 



= -2Vb\V- ^^^^x • VJ X (1 ~ cosl^) + sm^ • — X smRJ 



xe.TrAA^.R C . IT bh .R. ir bh R") 



= " 276^ Sin Y . — X sm-g X | sm-^ • — X sm ^ + cos ^ • - X cos ^ j 



2\e . irbh . R /'jtbh R\ 



= -Tys^^-3:7-*sin2--cos (^XT " W ' 



