11 
very great and b very small, so that we no longer have 
the impression of black lines on a white surface, but see a 
uniform grey surface, then the expression for its degree of 
whiteness will be /x( A — nbl) or W 0 (l — nr) if r denote^- 
Sometimes an engraver, instead of using parallel lines 
only, crosses the lines (cross hatching). With a given number 
of lines the tint will not be the same if he draws them all 
parallel, and if half are drawn at right angles to the others. 
Suppose we have 2 n lines, if drawn parallel the degree of 
whiteness will be W 0 (l - 2 nr), but let n of these lines be 
drawn perpendicular to the remaining n. Take the case of 
one of these perpendiculars, it will intersect one of the first 
series in a square whose area is b 2 , and as it is cut by n lines 
the sum of these will be nb 2 ; the additional white area, 
blotted out by this line, will be lb - nb 2 , and since there are 
n such lines the total area they blacken will be n(lb- nb*). 
Hence the remaining white area will be A - nib - {nib - nb 2 , 
which may be written A(1 - nr)*, since l 2 = A. If Wi denote 
the whiteness when the lines are parallel, and W 2 when 
they cross, we shall have 
Wi 1 - 2nr 
W 2 ~ (l -nr) 2 
In both cases the lines are supposed to be so thin as to be 
individually imperceptible. 
Again, suppose an engraver to cover a square white area 
with black circular spots which touch one another, the spots 
being very numerous and individually imperceptible, so that 
we receive the impression of a grey surface. If beyond this 
stage he exercised his skill in diminishing the area of the 
spots infinitesimally, and increasing their number, so that 
they still fulfil the condition of touching, it will make no 
difference in the intensity of the tint : for the area of the 
spots is always the same, and equal to that of the circle that 
can be inscribed in the square. 
