20 Mr. O-BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT 



(22.) Lasth', we shall try whether any other hypothesis leads to Fresnel's formulae in this case. 



If Fresnel's formulas hold, it is easy to see that 



s(a_a) = «'a', and therefore s {V - V) = s' V ; 

 therefore by (8) we have V^p^ = V"p", and therefore by (7) and (5) 



v./ = v"^, 

 from which it is evident that no other hypothesis except those in Articles (20) and (21) will lead 

 to Fresnel's formulse in this case. 



The second hypothesis here employed seems to me to be the only one we can adopt; for 

 it is extremely difficult to conceive how normal vibrations could be propagated more slowly than 

 transversal in a stable medium. 



If we do not suppose that V2 is very nearly equal to v", we may proceed to find F and V in 

 terms of V in the following manner. 



Substitute for V and V^ their values got from (6) and (7) in the equations (8) and (9) ; then, 



putting -r, = v, we have 



(^s-s')V- (m« + s') F = F" (.p, - p"), iiip - p') ( F + F ) = F" («" - ,s,). 



vp. — p" , ,. , 



Hence, if for brevity we put — (yup - p) = t], we have 



•^ S — l/Sj 



(,.«-«')f-(ms + o»'/ = v(j'+ y), 



and tijerefore V = ; F, 



fis + S +r) 



V +V 

 and from this expression we may easily find V, since F = '-. 



Since p = up, and P2 = vp", we have 



/ /' 



,; = iv' - 1) ((.^ - 1) . 4^ 



SECTION IV. 

 Explanation of the case of Total Internal Reflexioit. 



(23.) Since p = \i.p , p' will be > 1 when p is > /x (which it may be when yu is < 1), 

 and then s will be impossible ; and Fresnel's formulae become imaginary ; which indicates that 

 the equations of connection cannot be satisfied by the three rays in Article (15), or the five rays 

 in Article (l9). We shall now consider how the equations of connection may be satisfied under 

 such circumstances, and first in the case of vibrations perpendicular to the plane of incidence. 



Let us suppose that the general value of F is 



a ,_ 



2 ^ ' 



It is allowable to give V this value, though it is imaginary, since it is an integral of the equa- 

 tions of motion, and is capable of satisfying, analytically, the equations of connection, and the 

 equations (A), (B), (C), Section i, (see Article 5). Moreover, by superposing two such imaginary 

 values of V, viz. ae*'"' "?''""''*'-' and oe"'*"'-''-")'^'-', we obtain a real value, viz. a cos Ic {vt-p,v-sx), 

 which will of course satisfy the same linear equations as the two expressions of which it is the sum, 

 i. e. the equations of motion and of connection. 



