BANDS SEEN IN THE SPECTRUM. 
241 
SO that^=0. Then the expression (17-) may be dispensed with, since it only holds 
good when g'=0, in which case it agrees with (16.). Let Tq be the value of the 
thickness T for which g'=0. ThenT=0 corresponds tog'= — (/z+Zr), T=To tog'=0, 
and T=2To to g’=h-\-h ; and for values of T equidistant from T^, the values of g' 
are equal in magnitude but of opposite signs. Hence, provided T be less than 2T„, 
there are dark and bright bands formed, the vividness of the bands being so much the 
greater as T is more nearly equal to Tg, for which particular value the minima are 
absolutely black. 
Secondly, suppose the breadths h, k of the two streams to be equal as before, but 
suppose the streams separated by an interval 2^; then the only difference is that 
g'-= — {h-\-k) corresponds to a positive value, Tg suppose, of T. If T be less than To, 
or greater than 2To — Tg, there are no bands; but if T lie between Tg and 2To — To 
bands are formed, which are most vivid when T=Tg, in which case the minima are 
perfectly black. 
Thirdly, suppose the breadths h, k of the interfering streams unequal, and suppose, 
as before, that the streams are separated by an interval 2g; then g'= — (A-f-A:) cor- 
responds to a positive value, Tg suppose, of T: g' z= —[h^k) corresponds to another 
positive value, T^ suppose, of T, T^ lying between T 2 and To, Tq being, as before, the 
value of T which gives g'=0. As T increases from Tg, g' becomes positive and increases 
from 0, and becomes equal to h->-k when T=2To— T^, and to h-\-k when T=i2Tg — T 2 . 
When T<T 2 there are no bands. As T increases to T^ bands become visible, and in- 
crease in vividness till T=Tp when the ratio of the minimum intensity to the maxi- 
mum becomes that of h—k to h-\-^k, or of k^h to according as h or k is the 
greater of the two, h, k. As T increases to 2Tg— the vividness of the bands 
remains unchanged ; and as T increases from 2Tg — to 2Tg— T 2 , the vividness 
decreases by the same steps as it before increased. When T 2rg — T 2 , the bands 
cease to exist, and no bands are formed for a greater value of T. 
Although in discussing the intensity of the band^s the aperture has been supposed 
to remain fixed, and the thickness of the plate to alter, it is evident that we might 
have supposed the thickness of the plate to remain the same and the aperture to 
alter. Since •urco T, the vividness of the bands, as measured by the ratio of the maxi- 
mum to the minimum intensity, will remain the same when T varies as the aperture. 
This consideration, combined with the previous discussion, renders unnecessary the 
discussion of the effect of altering the aperture. It will be observed, that, as a general 
rule, fine bands require a comparatively broad aperture in order that they may be 
well-formed, while broad bands require a narrow aperture. 
16. The particular thickness Tg may be conveniently called the best thickness. 
This term is to a certain extent conventional, since when h and k are unequal the 
thickness may range from T^ to 2Tg — without any change being produced in the 
vividness of the bands. The best thickness is determined by the equation 
- ^|=^(4g+/i+/0- 
