Prism on Newton's Rings, 347 
where b depends upon the curvatures. The black of the 
nth order for wave-length X occurs when 
i*X = <* + &£*, (3) 
or 
* = v^{(M-«)/6}, (4) 
so that 
^ = i n (v 
The wth band, formed actually at a?, is seen displaced under 
the action of the prism. The amount of the linear displace- 
ment (£) is proportional to the distance D at which the 
prism is held, so that we may take approximately 
/3 representing the dispersive power of the prism, or grating. 
The condition that the nth. band ma}' be achromatic (for 
small variations of \) is accordingly 
*?-». m 
or 
1 n 2 
i6Wm = i)lX ~ a ' ' * * * • (8) 
a quadratic in n. The roots of the quadratic are real, if 
/3 2 D 2 b > a/X 2 (9) 
If a be zero, the condition (9) is satisfied for all values of D, 
so that at whatever distance the prism be held there is 
always an achromatic band. And if a be finite, the con- 
dition can still always be satisfied if the prism be drawn 
back far enough. 
From (8) if n 1? n 2 be the roots, 
'± + ± = £ (10) 
ra x n 2 la v J 
Again, if a = 0, that is if the plates be in contact, fti=0, and 
n 2 = 8\p 2 ~D 2 b\ (11) 
The order of the achromatic band increases with the dis- 
persive power of the prism and with the distance at which 
it is held. The corresponding value of x from (4) is 
# = 2\£D (12) 
