IN THE NEIGHBOURHOOD OF A CAUSTIC. S8^ 



because every positive value is balanced by an equal negative value ; 

 and therefore the expression for the disturbance of ether at the 

 illuminated point is 



sm — {vt- E) j.- cos — . jpY^ (s! * -h-^U 



the integral being taken between — infinity, + infinity ; 



or, 2sin — (vt-E) j.^cos "^-g^p^I^* .^p.ss), 



the integral being taken from to infinity. 



Making "T" • ^ p y-3 • ^" = ^^'' <"■*'= -f-^ (5-) • "'. and putting m 



for Sp X (-j-\ , and omitting the constant factor, we find as the ex- 

 pression for the disturbance of ether at the illuminated point 



sin — {vt — E) j„ cos- {vf — m. w), 



and therefore the expression for the intensity of light is 



[X,cos ^ {w' - m.w)Y, 



the integral being taken from jt = to w = infinity. 



It will be observed that m is proportional to ^p, and therefore the 

 intensity of light at the Geometrical Caustic, or where Sp = 0, is found 

 by making w = in this formula. 



16. In the second case, the expression for the whole disturbance 

 of ether is 



/..sin^{..-F-3L(,._^^^..'){. 

 which, as above, is shewn to be equal to 



2sm-(.^-i^)./.cos-.3^(x'-^;,.^). 



from a:' = to x = infinity. 



3 D 2 



