344 Mr KELLAND, ON THE TRANSMISSION OF LIGHT 



We shall find 



cos 9i = sin e sin \//, 



cos <pi= — cos e sin >/a, 



cos \j/i = cos \//, 



cos 02 = cos IX cos € — sin n sin e cos ■>//, 



cos <p-i = cos f). sin e + sin ^ cos e cos v//, 



cos \|/2 = sin /i sin ^1/, 



cos 03 = — sin Hi cos c — cos ,u sin e cos ^j/, 



cos (p-i= — sin M sin 6 + cos /i cos e cos >/^, 



cos ^i = cos M sin \//. 



19. By these values of the angles, the equation in (17) is reduced 

 to 



— 3/ (cos n cos e — sin M sin e cos ■^) (sin jn bos e + cos ^ sin e cos \//) 



— iV(cos M sin e + sin n cos e cos >//) (sin m sin e — cos ^i cos e cos \//^) 

 + /• sin Hi cos fjL sin^ >|' = 0, 



or — ilf {sin a^i (cos^ e - sin" e cos^ ^) + cos 2m sin 2 e cos >//} 

 — iV |sin 2ix (sirf e - cos° e cos- \|/) — cos 2^ sin 2£ cos \//} 

 + P sin 2/u sin' ^ = 0, 



or sin 2/ui {(M- N) (cos= e - sin'' e cos' v|/) + (N- P) sin' >/.} 

 + cos 2ix {M— N) sin 2 e cos >// = 0. 



20. The same substitutions reduce the value of v' to 



v^ = rt' cos' 02 + V cos' ^2 + c' cos' >//£ 

 = a' (cos' |ii . cos' e + sin^ ^ sin' e cos' >//— ^ sin 2/u sin 2e cos \//) 

 + J' (cos' M sin' £ + sin^ n cos'^ e cos' \// + ^ sin 2|ii sin 2£ cos \//) 

 + <? sin' M sin' >// ; 



