646 Mr. stokes, on THE FORMATION OF THE CENTRAL SPOT 



refracted normal waves are replaced by undulations of the kind which I have called superficial. 

 Now the existence of these superficial undulations does not affect the interpretation which has been 

 given to the expressions (J) when the angle of incidence becomes greater than the critical angle 

 corresponding to the refracted transversal wave ; in fact, so far as regards that interpretation, it 

 is immaterial whether the expressions {A) satisfy the linear equations of motion and condition 

 alone, or in conjunction with other terms referring to the normal waves, or rather to the superficial 

 undulations which are their representatives. The expressions (D) however will nol represent the 

 whole of the disturbance in the two media, but only that part of it which relates to the transversal 

 waves, and to the superficial undulation which is the representative of the refracted transversal wave. 



6. Suppose now that in the expressions {J) n becomes imaginary, n remaining real, or that 

 n and n both become imaginary. The former case occurs in the theory of Newton's Rings when 

 the angle of incidence on the surface of the second medium becomes greater than the critical angle, 

 and we are considering the superficial undulation incident on the third medium : the latter case 

 would occur if the third medium as well as the second were of lower refractive power than the 

 first, and the angle of incidence on the surface of the second were greater than either of the critical 

 ano-les corresponding to refraction out of the first into the second, or out of the first into the third. 

 Consider the case in which n becomes imaginary, n remaining real ; and let y/ p _ i = y. Then 

 it may be shewn as before that we must put - vs/ - \, and not v \/ —\, for n ; and using p, Q 

 in the same sense as before, we get the symbolical system, 



(E), 



to which corresponds the real system 



V = pe-''-'cos{k{vt -Lt) - 9], 



V = p/" cos {k(vt - lac) - 6,], f (•F'). 



r= p'cos\k'iv't-l'a!-n'!i!)-e'\, 



When the vibrations take place in the plane of incidence, T and F in these expressions must 

 be interpreted in the same way as before. As far as regards the incident and reflected superficial 

 undulations, the particles of ether in the first medium will describe small ellipses lying in the plane 

 of incidence. The ellipses will be similar and similarly situated in the two cases ; but the direction 

 of revolution will be in the case of the incident undulation the same as that in which the refracted 

 ray would have to turn in order to diminish the angle of refraction, whereas in the reflected 

 undulation it will be the opposite. 



It is unnecessary to write down the formulaj which apply to the case in which n and n' both 

 become imaginary. 



7. If we choose to employ real expressions, such as (D) and (F), we have this general rule. 

 When any one of the undulations, incident, reflected, or refracted, becomes superficial, remove z 

 from under the circular function, and insert the exponential e"*", e*"", ore"'"", according as the 

 incident, reflected, or refracted undulation is considered. At the same time put the coefficients, 

 which become imaginary, under the form p (cos 6 ± \/- I sin 6), the double sign corresponding 

 to the substitution of ± i/-\/- 1, or ± y' v/ - 1 for ra or w', retain the modulus p for coefficient, 

 and subtract 6 from the phase. 



It will however be far more convenient to employ symbolical expressions such as (B). These 

 expressions will remain applicable without any change when n or n becomes imaginary : it will 



