On the Resolving Power of a Spectroscope. 763 
<f). The angular width of the image resulting therefrom is 
given by A0 where 
A0 _ cos 6 
A<f> ~~ cos 0' 
It must be observed that whether this decreases or not as <$> 
decreases can only be decided when it is known which side 
of the normal the image is on. 
Two spectral lines for which the wave-lengths differ by 
S'X will be at an angular distance apart S0 where 
B0_ n 
8\ e cos ' 
In these formulae, A0 is a measure of the broadness of each 
image and 80 is the separation between the centres of neigh- 
bouring images (or dispersion). 
Now, in estimating the purity of the spectrum, it is clearly 
the ratio of the second to the first of these two quantities 
which is important. Greater broadening of the image would 
be consistent with greater purity if the dispersion was 
increased in proportionately greater amount. The ratio of 
S0 A0 
to — 
8\ A</> ' 
a ratio which may be spoken of as the tilt-advantage, is 
A= 
e coscp ' 
whence it is seen that the advantage of the arrangement 
increases as (/> increases, whether the spectrum examined is on 
the same side of the normal as the incident beam or not. The 
relative advantage compared with the case of normal incidence 
is 1/cos <j>. 
In illustration of the meaning of this result, it may be 
pointed out that the case in which greatest separation between 
the centres of the images is obtained is not the case in which 
greatest advantage is gained. Greatest separation between 
the centres occurs when cos = ; that is when 
e(l + sin cp)=n\. 
This case can be obtained by starting from normal incidence, 
selecting any spectrum and turning the grating so as to 
increase numerically. The relative advantage in this case 
can be only moderate. On the contrary, in order to increase 
the angle of incidence to 90°, the grating must be turned the 
other way ; in this case greater purity is obtained although 
