INTENSITY OF REFLECTED AND REFRACTED LIGHT. 403 



and if we assume 



'y= a cos [ex -\-fy + c #) + 5 cos {^ — ex ■\-fy ■\-ct-\-g) 



7/ = a, cos {e, x, +fy + c ^ + /*) 

 then ^7= a COS (ex-\-e8 x+fy + ct + f8y)—aco&ex+fy + ct 



-\-b<ios( — ex — ebx +fy + ct+g +/d y) — b cos ( — ex +/y + ct + g) 



= —acos{ex+/y + ct) {1 — cose 8 x+/8y) 

 — asm{ex+/y + ct)sia(e8x+fdy) + etc. 



Let e X +/y be abbreviated by p 

 ex-/y R 



e^x+fg p. 



and ^7= — acos^ + c^(l — cos^^) — asinp + c^sin ^^ 



— icos( — R + c^+^)(l — cos^R) + 6sm( — R + cif + ^)sin^R 



= -I.2sm2-^ -R2sin«-^ 



\ d\ . ^ 1 c?R . s^T^ 



+ -^— smop + -j— smoR 

 e ax ^ e dx 



denoting 7 by I + R. 



Now the wave is similarly situated with respect to the line along which R is 

 measured, and that along which g is measured ; hence 



^80 / , d)V cv A . . ^R 



2(,^. + ^a.'^)sin!|£=2(0. + ^" 8z^) 



sm^ 



which gives each of them =2" 



For g'=-2 (0. + ^^.-)2sin^-|?-.7 



&c. = &c. 



The equation corresponding to (3j is 



and 7, —'y—a,cos{e,x-]-e,8x+fy+f8yJrCt + h) 



— a cos (e X +fy + ct) — h cos { — ex +fy + ct+g) 

 = 7,cos^p,H P-'sin^p,— 7 



