18G UNDULATORY THEORY OF LIGHT. 



confliGt with the HclvaDciug wave, which it meets at the first surface. This 

 advancing wave does not change its phase by refraction, but the reflected part 

 of the return wave does so, and is therefore in harmony wnth the advancing 

 wave Avhich it joins. The two accordingly conspire from tliat time forward; 

 their emergent portions at the second surface producing a bright ring by trans- 

 mitted light, while their reflected portions, returning to the firtt surfiice, conflict 

 with the next advancing wave which they meet there. 



But at the points where the bright reflected rings appear, the case is different. 

 The return wave is in harmony with the advancing w^ave which it meets at the 

 first surface, and its emergent part conspires with the reflected part of the 

 advancing wave. But its reflected part, losing half an undulation, conflicts with 

 the transmitted part of the advancing wave, and thus produces, by subsequent 

 transmission through the second surface, a ring parti.dly obscure, but not entirely 

 so, from the great inequality of the conflicting molecular velocities. If avc 

 disregard the successive advancing weaves, and consider the successive values of 

 the terms m, f'u, i^u, &c., at these points, ]n-oceediug from a single original 

 Avave, we shall fuid them alternately positive and negative. Their emergent parts 

 must be so likewise. And since they are decreasing, their sum takes the sign 

 of the first term, which is positive ; so that their resultant conspires with the 

 wave reflected to the eye from the first surface. The components, simultaneously 

 reflected to the second surface from the -first, form a similar series with signs 

 reversed, and therefore have a negative resultant, conflicting Avitli the wave 

 emcr":ent at that surfiice.* 



* This matter may perhaps be made more clear as follows : 



Callinf^, as above, u the value of the molecular movement in the ray transmHted through 

 the first surface, and v the ratio of reflected to incident light, the advancing and returning 

 waves within the lamina will have the successive molecular velocities — 



1. Advancing wave, ?/, t-it, v^u, v'''u, v^n, &c. 



2. Returning wave, vn, r^it, v-'ii, v'u, &c. 



And the squares of the velocities of the emergent comijonents will be — 



til' til' tti' lU 



1st surface, — (v-ii- — f'w-), — (v^'ii- — tfu-), — (r"'H' — c'-h-), — (r'-'if- — v^'^'u"), &c. 

 in 111 rn m 



n-i-ij ?// ??t 'ill 



2d surface, — vu" — r-«-1, — (i:''«- — 1""«-), —i'c^u- — ?;'"«-), — {v^-iC- — c'-tw-^), &c. 

 m '• m m m 



Consider the movement in the incident -wave to be positive. Then if the lamina were 

 without thickness, the successive reflections still going on, the sign of the movement in the 

 diminished waves successively emergent would be always positive for the second surface, 

 (for which the number of reversals by reflection is always even, ) and always negative for the 

 first, (for whiclr the number of similar reversals is always odd.) 



By giving greater or less thickness to the lamina, any diliereuce of path may be introduced 

 for cither the rays seen by reflection or those seen by transmission. ]f 6 represent the thick- 

 ness, the differences of path which Aviil exist, after the several successive reflections, will be 

 2d, W, (if, or generally Mm6, m being any integral number. If (J=7tXP-, or 20=2nXi'^=n'A, 

 n being also any integral number, it is manifest that 2in6=mn?,, being a number of complete 

 undulations, cannot change the sign of the movement, whether m be even or odd. 



But if 2(/=(-2«-|-])X-J-A, then 2md=m(2n-\-l) Xp^ will be an odd number of half undula- 

 tions when m is odd, and an even number of half undulations, or an integral number of 

 complete undirlations, when m is even. Accordingly, the wave changes its sign for every 

 odd value of m. 



Hence, if 2()=ii?.; or 0=nXi-^, the movement will be negative for all component waves 

 emergent from the first surface, and positive for all emergent from the second. 



Also, for 20=^{2)i-\-\)Xi/^, or (?=(2»j+1)XJ^, the signs will be alternately positive and 

 negative at the first surface, and negative and positive at the second. 



For the first case, if we take the square roots of the scpiares of the' emergent components 

 given above, substituting the value of u, Ave shall have for the resultant v' at the first surface, 



v'={'i}- — l).(v-]-c^-\-v'^ ad inf.) Whence v'= — v. 



The interference is therefore absolute, and the rings formed at these thicknesses will be per- 

 fectly darli. 



For the rays emergent at the second surface Ave obtain the expression — 



u'={l—v-).{i+v--{-v^....adiiif.)={l—v-).^^-^=l. 



