412 PROFESSOR KELLAND ON FRESNEL'S FORMULA FOR THE 



= 2 !-'- ( S ijl- - 8 .r/-) sin </)' cos 2(/>' X j - 2 T sin'^ "2-' + - -^ sin K X, j 



3(^^ . , „, 3M, clT . ,, ,,, 

 rr — 2" T sm ^' cos,-(p' + J- Sin (^ cos '(p 



By adding this term to that just found, we get 



2 IVT /■/ T* 



Q, (a -a)-|- T sin (^' + -^ -^ sin ^'-D,T, 



Hence 



I , , U . . /d\ dR\ M, . ..dT 



^'' !l_R cm rfi 4. T sin rf)'! + O fa — a'>4 



^.'- ^! , , U . ^ /d\ dR\ M, . .dT ^, ^^ 



^1-=-! jr::Rsin0 + Tsin<^'| +Q.(a,-«) + ysm<^(^-^)+-sm<^'^-Dl,-D,T, 



14. From this equation we obtain, by interchanging the quantities (1- R) sin <p, 

 Tsin0'&c. 



^= -^ (f:^ sin ^ + T sin 0') 

 dt' 2 



-D,T,-DI, 



By subtraction 



d' a (P a, 



= -(Q.+ Q)(a-a,) 



Now Q,+Q differs from c' by a finite quantity: hence this equation can only be. 

 satisfied by making 



d^a., (^^ a _ p. 



the second of which equations is a consequence of the first. 

 By adding the two equations we get 



d- a (7- a. 



- -r + -r^= -c'^ (_l-E sin (b + T sin <p') 

 at dt- 



+ (Q-Q,)(«-«,)-2D.l,-2D,T, 

 2U . _, (d\ dR\ 2M, dl 



