MR. G. A. SCHOTT ON THE REFLECTION AND REFRACTION OF LIGHT. 881 
film must lie between the above limits, this is only a sufficient, not a necessary 
condition. 
There is another point to be considered, the magnitude of 27rr//\. It was shown 
(p. 857) that if be less than the reciprocal of the greatest value of p for the film, 
the expressions found will be conveig'ent. It follow^s that j D | < 
^ ° ' p max. pp-^pij“ 
E I < --■ 
A + Poll'd (pf -r Po") 
p max. 
These are the limits which the absolute values of D, E must not exceed. Consider, 
for instance, the case of water-flint-glass, for which Pi= 1 ’336, Pq= 1 ‘616, | E [ = ’075. 
If p lies between p^ and pg, then p max. = pg = 1’616 ; the greatest numerical 
value of A — pg^ — pi^ -b Gpg^p^^ occurs for p = \/(Ai|#i) throughout, and is (pj — p,,)® 
(see p. 880) or ('28)^. The greatest value of p^^ a is given by p = pg and is therefore 
Po^ — pi^. Hence we must have 
D 
2 X 1-616 
< - < 2 
1-616 
E I < 2-? 
1-616 (1-616)- - (1-336) 
<9x <-190 
If p > Pg, p max, = 2-67 (about the greatest value of p known for a transparent 
substance), the maximum of pj^ Ais(2-G7)^—(l'33G)^that of A — pg^ — pp-|-Gpg-pj^ 
is \ 2-67 - 
(L616)2 
2-67 
2-67 — 
(1-336)2 
2-67 
Hence 
D I < 
X 1-616 (2-67)2 - (1-336)2 
2-67 ' ' (1-616)2 - (1-336)2 
< 
o. o O 
o JoJ 
5-35 
2-67 0-83 
< 7-74 ; 
E 1 < 
X 1-616 {(2-67)2 - (1-616)2} {(2-67)2 - (1-336)2} 
2-67 
(2-67)2. {(l-616)2-(l-336)2} 
< 
3-232 
2-67 
4-52 X 5-35 
7-13 X 0-83 
< 5-00. 
These expressions show that it is possible (or at least probable) to satisfy the 
conditions for convergence by conceivable values of p for the layer. And since these 
upper limits for | D | , | E [ are much wider on the second supposition, and rather 
close to the actual values of ] D | 1 E [ on the first supposition, there is a very 
distinct presumption in favour of the second, namely, that the average value of p for 
the variable layer is greater than 1‘616 (and less than 2-67). Quincke does not state 
whether his reflecting surfaces were polished with ferric oxide or not, but this is a 
common enough material. Emery also has a higher refi-active index than I'GIG, 
so also diamond dust, and some one of these would })erhaps be used, clialk or silica 
being hardly hard enough for the pui-pose. 
£ 
* The retardation of phase of light polarized in the plane of incidence is tan~i (U cos <7). Werkicke 
finds that this retardation is at most a few thousandths of a wave-length, so that D is probably less than 
-QI, and quite incapable of reasonably accurate measurement. 
MDCCCXCIV.—A. 5 U 
