ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 
685 
there is no duplicity, and in iii. only an attempt, while in ii. the 
separation may be regarded as complete. For case ii. the same result 
follows, even when the points or lines are twice as close as before, for 
in this case the resultant amplitude is 
sin u siu (u -f- J tt) 
U U -j- \ TT 5 
and this vanishes when u = — \ tt. The maximum value of the resultant 
amplitude which takes place at a point near u = ^ 7r is much less 
than before. The image, in fact, very incompletely represents the 
object ; “ but if the formation of a black line in the centre of the 
pattern be supposed to constitute resolution, then resolution occurs 
at all degrees of closeness.” The author has illustrated these results 
experimentally by the observation through a telescope of two parallel 
slits in films of tinfoil or silver. The distance is chosen so that when 
backed by a neighbouring flame the double part of the slit is just mani- 
fested by a faint shadow. On replacing the flame by sunlight through a 
distant vertical slit, w'hen everything is in line no sign of resolution of 
the double part of the slit is observed. A slight sideways movement of 
the telescope then suffices to bring in the half-period retardation, and a 
black bar down the centre is at once seen. In accordance with theory, 
this blafck bar is still seen when the distance is increased much beyond 
that at which duplicity disappears under flame illumination. 
The calculations for a circular instead of a rectangular aperture in the 
case of a double point lead to similar results in the three cases i., ii., iii., 
as before, except that the jmrtial separation, indicated by the central 
depression in curve iii. (fig. 106) is here lost. 
The author then extends the calculation from the consideration of a 
double point or line to the case where the series of points or lines is 
infinite, constituting a row of points or a grating. First taking the case 
where the various centres radiate independently, as if self-luminous, if 
the geometrical images are situated at n = 0, u = ± v, u =■ ± 2 v, &c., 
by the preceding the expressions for the intensity at any point u may 
be written as an infinite series, 
i oo 
sin 2 (to -f- v ) f sin 2 (u — v ) 
v2 r ' 
( u + vy 
(u — vy 
+ 
sin 2 ( u + 2 sin 2 (u — 2 v) 
+ 2 vf + (u - 2 v) 2 + * 
which may be expanded by Fourier’s theorem in a series of cosines. 
Thus 
I (w) = I 0 + Ii cos 
2 7T u 
V 
+ . . . . 
_ 2 7r r u 
-f- lr COS 
V 
in which, as shown by calculation, 
according as v exceeds or falls short of r tt. 
