1898-99.] Lord Kelvin on Reflexion and Refraction, 375 
By these equations eliminating I / and J from (22), we find 
— ( bp j - b t p ) I + ( bp t + b t p) V = a(p / - p)J' 
' P) 1. 
a (P, “ P) (I + I') = “ ( G Pj + G jP)^' i 
and solving these equations for I' and J' in terms of I, we have 
T' - ( 7; P- “ h fi) ( e Pi + c fi) ~ ® 2 (p, ~ P)1 l 1 
(27); 
(bp t + b t p) (cp / + e t p) + a 2 (p / - p) 2 
(28), 
I 
j' = ~ 2 abp / (p j - p) ^ 
(b Pl + b /P ) (c Pt + c,p) + a 2 ( Pj - p) 2 x j 
and with J' and V thus determined, (26) give J, and I /, com- 
pleting the solution of our problem. 
§ 13. Using (18) to eliminate a , b, b t , c, and c y , from (28), and 
putting 
Wmm I./, 
find 
and 
p J cot j + p cot j t 
I ' _p, cot i- p cot i / - h(p / - p) 
I p j cot i + p cot i t -l- h(p / - p) 
J y _ - 2iip t cot i 
(29) ; 
(30) , 
(31) . 
I p y cot i + p cot i t + h(p t - p) 
Consider now the case of v and v t very small in comparison with 
u and u t ; which by (28) makes 
cotyjgj Uva, and cotj = l/v f a . . . (32). 
This gives 
h= (P- ~ P) sm 1 (33) 
u u 
Pj - + P ~ 
V V, 
which is a very small numeric. Hence J' is very small in com- 
parison with I ; and 
I' . p y cot i — p cot i j 
I ~p J cot ^ + p cot ^ 
§ 14. If the rigidities of the two mediums are equal, we have 
p y | p = sin 2 i | sin 2 i /t and (34) becomes 
I' _ sin 2 i - sin 2 i t _ tan (i - i) 
I sin 2 i + sin 2 i t tan ( i + i 
which is Fresnel’s “tangent-formula.” On the other hand, if the 
densities are equal, (34) becomes 
I' _ - sin {i - i) 
I sin (i + i) 
( 36 ), 
