178) Diffraction-Images formed by Plane Diffraction- Grating. 
to be exceedingly small, yet, in view of the fact that measure- 
ments of spectra are now made with extreme accuracy, it 18 
necessary to become assured that the usual treatment has 
the accuracy required. 
We shall write, Mas necessary, w=pv where p=7/(7 +3), 
AU 
and consequently — =p. 
Then 
a 1 
oA == 2y* fn cot nv—cote+ p( cot u— a; ; 
Now the principal maxima occur for values of v equal or 
very nearly equal to pa where p is an integer; writing 
v=pr +a we obtain 
fi) eae ees | ne —Il 1 ’ 
 Mlhaes Lanier 3 a+ p( cot u— ') { 
approximately, « being small. 
Now, in practice, we are only concerned with those cases 
in which sin w is considerable ; for in other cases the spectra 
are necessarily weak. If we confine attention to those cases 
in which the intensity is not less than one-fourth the maximum 
possible, w must not differ by more than 60° on either side of 
an odd multiple of 7/2. Hence cot u lies between +4/8 and 
—,/3, and the maxima will be given by values of « such that 
ote 4. : $ 1 
n'—l, is not numerically greater than p /3+ —: whence 
‘ - DIT 
a is not greater than - 
Importance attaches not so much to the actual value of @ 
as to the relation it bears to the least detectable value of 
it : | ’ 
dn/X, viz: —— where np is the “ resolving power” of the 
yyre 1 ry np 
grating. 
Now dry dv a 
Pai 4 ~ pt 
. Ad i) 1 
which is less than -—, and this bears to the least 
pT 
~ an . 
detectable value of ree the ratio 
BP ; 6 
oe oe, hor practi rally- 
np / np " VT" 
Therefore the error committed in the usual method of ealcu- 
lating the positions of the principal maxima is far below the 
least detectable error provided » is considerable ands‘ the 
maximum considered is fairly bright. 
