185 
of Edinburgh, Session 1869 - 70 . 
will contain nothing but a series of equal and opposite phases, 
which will be discordant and mutually destructive, as far as con¬ 
cerns the propagation of light in the direction AP, and no image 
will be seen in that direction, whatever may be the distance be¬ 
tween the transparent spaces. The same will be the case if the 
breadth of the spaces be such that the surface AC stretches across 
exactly 2, 3, or any whole number of wave lengths. But if the 
surface AC stretches across n + ^ wave lengths, ^ being a proper 
fraction, the vibratory movement transmitted along AP by the 
fractional part of the wave length will not be destroyed by the 
concurrence of its complete opposite, and light will be propagated 
along AP. The other transparent spaces will send concordant 
phases to the envelope wave, if AP be at the proper angle. In 
this case, however, the breadth e of a transparent space must be 
added to a in the formula a sin D, &c., a + e being the distance 
between the successive effective remnants of the vibratory move¬ 
ments which pass to the envelope surfaces. The breadth a 4- e 
occupied by a dark and a transparent space is called an element of 
the grating. If the fractional part of the wave length, which is 
effective in forming any one of these envelope waves, be either a 
very small or a very large fraction, its effect will be feeble, and 
the corresponding image of small intensity; but if it be exactly 
one-half of the wave length, its effect will be the greatest possible, 
and the envelope wave will receive from each opening the greatest 
possible amount of concordant action. In this manner is explained 
the difference of intensity of these lateral images, the one nearest 
to the central image not being always the brightest. Proximity 
to the central image is, however, also a cause of greater brightness, 
it being evident that the less the surface AC in the last figure is 
inclined to the incident waves, the greater is the absolute length 
of that part of it which stretches over any given fraction of the 
wave length, and the greater the amount of action of which it is 
the locus. 
In the above the incident waves have been supposed to be ex¬ 
actly parallel to the plane of the grating, so that the same phase 
of vibration passes at the same instant through all the openings. 
