BANDS SEEN IN THE SPECTRUM. 
239 
=|, from gi= — {h,+k,) to g,= — {h,^k,) 
=0, from g^= ~{h,^k;) to g^=-\-{h^^k) 
= -|, from g,=h,^k, to g,=h,+k, 
=0, from gi=hi-{-ki to g,=^ . 
Now vanishes when g^ is infinite, on account of the fluctuation of the factor cos^^;? 
under the integral sign, whence we get by integrating the value of ^ given above, 
and correcting the integral so as to vanish for co , 
m;=0, from g,=:—co to ; 
u’= from g,= - {h,+k,) to g,= - (V^/) ; 
w=<7rk^ or (according as h^>k^ or h^<k^,) from g=. — (Ji^^k^ to ; 
^=^{ht+k—g), from gi=h'^k^ to 
u;=0, from + to ^y=oo . 
Substituting in the expression for the intensity, and putting g,= 
-f2, SO that 
j— 
'SjXf , , 
— — 4g-h-k, 
(14.) 
we get 
1 = ^(h+k), 
when the numerical value of g' exceeds h-\~k; 
(15.) 
1= \/^'^) cosf'j, . . . 
when the numerical value ofg' lies between h-\-k and h^k; 
OR/ 2Ti/ 
1= {h-{-k-{-2hcos §'), or = -jr {k-\-k-\-2k cos §'), 
(16.) 
(17.) 
according as ^ or A; is the smaller of the two, when the numerical value of g' is less 
than h'k. 
The discontinuity of the law of intensity is very remarkable. 
By supposing g,=0, ki=hi in the expression for w, and observing that these sup- 
^ dp 
positions reduce w to J s\n^ h^p we get 
a result already employed. This result would of course have been obtained more 
readily by differentiating with respect to A,. 
