AN APPARENT NEW POLARITY IN LIGHT. 
229 
sin 
sin T {{7~ e (y + v) 2 ~ R } 
+ cos x (v t -f)f y sm { ^ (y + v)‘ ~ R } 
in x (v t -/) [ cos R ,f y cos ^ (y + x)" + sin R X sin T7i{y + tY } 
+ cos t * -/) { cos R -X sin ^7 {y + t)~ ~ sin R X cos wi O' + v) 2 } ’ 
where the limits of y in the integral must be the values which determine the extent 
of that portion of the great wave which is affected by the interposition of the refract- 
ing substance. 
Suppose now that a plate of mica or other refracting substance is so placed as to 
cover a portion of the pupil on that side on which y is considered positive ; and sup- 
pose that its limits, measured in the same way, are from y = + g to y indefinitely 
great : suppose also that the limits of the pupil are from y — — h to y — -j- li ; then 
the integral which is independent of R must be taken from y — — h toy — + g, and 
that which depends on R must be taken from y = g to y = -f h, and the integrals 
must be added to obtain the whole disturbance of ether at the point of the retina. 
Our expressions then depend entirely on the two integrals^* cos (y + c -^ 
and J* sin (3/ + . For estimating these, we shall refer to Fresnel’s inva- 
luable table of/*cos ~ s 2 and^^* sin-^- s 2 *. If s — \/ yyj • (y -f- it is easily 
seen that J y cos (y + ^)“ = \J cos s 2 , the limiting values of s being 
found from those of y by the equation s — \J XcV {y + Omitting the con- 
stant multiplier - (remarking only that, as it becomes infinite when a — 0, and 
impossible when a is negative, different investigations must then be used) ; and 
putting C (s) for^* cos ~ s 2 and S ( s ) for J] sin-|- s 2 ; and remarking that, as Fres- 
nel’s integrals vanish when s = 0, and as the differential coefficients have the same 
values with the same sign for negative and positive values of s, the integrals must 
have equal values but with opposite signs for negative and positive values of s, or 
that C (— s) = — C (.s), S (— s) = — S (s) ; we find that the integral, in the case 
before us, may be represented as follows : 
* Mdmoire sur la Diffraction, Mem. de l’lnstitut, 1821 and 1822 . 
