Manchester Memoirs, Vol. Ixiii. (1919) No. 4 5 



1^5. We proceed to discuss the results of §4, when Q is small. When 

 this is the case, the parallelograms typified by ABCD have their sides 

 AB and BC elongated (see equations (/V) ) ; whilst assuming that 

 {a^-^b^ is a multiple of (fli + ^i), the distance AA^^^ is given by {iv). 

 But {iv) may be written 



/i = ^^n+i = 2« (^i + ^i) cosec Q sin {e\2) = n{a, + b,) sec {e\2) . {vi) 



Since AC approaches the value AB + BC={a^ + b^ + a^ + b^) cosec 

 ^ = («+:) (a^ + z^j) cosec f^, as d diminishes; it follows that with 

 diminishing values of 6, the ratio ^^„+i / AC approaches the limit 

 2n sin (el 2) I («+i). Thus the ratio of the pitch AAn+i of the 

 patches along the row AAn+i to the length of a patch diminishes 

 indefinitely with 0. It may be noticed that (vi) approaches the limit 

 « (^i + ^,) = a2 + ^2. We can sum up these results by saying that the 

 pattern B^ resolves itself into transparent patches arranged in rows 

 parallel to AAn+i, the pitch p, in the direction AA^+i approaching 

 the invariable value (a^ + b^), whilst the length of each patch {i.e., AC) 

 increases according to the cosec law. 



The direction AA^+i clearly bisects the angle A^AA^\ i.e., it makes 

 an angle (tt + 0) / 2 with AB. This is shown in Big. 2. Running 

 parallel to the row of transparent patches given by AA^j^^, we shall 

 have a parallel row of patches given by KA^. This row will be 

 distant 



(ai + ^i)cosec^cos(^/2) = (ai + 3i)/2sin (^/2) . {vii) 



from AAn+i, i.e., this is the perpendicular distance between AAn+i 

 and J^An. Obviously as d becomes small, the ratio of /a to/i rapidly 

 increases. In the particular case when Oz + bz — a^ + b^, i.e., « = i we 

 have 



/, := 2(ai + b^) sin (6'/2) cosec $ = {a, + b^) sec {BI2) . {viiia) 

 p-2 = ("^i + bi) cos {6\2) cosec 6 = ^{a, + b^) cosec (6*/2) . {viiib) 



§6. We now come to operation (/;') of §3, i.e., we have to discuss 

 the effect of superposing the sets of bands Y^ and Y^- But Y^ is 

 exactly the same as X^ except that its bands are at right angles to those 

 of X^. Similarly Y^ is the same as X^, except that its bands are at 

 right aiagles to X^. Thus Qi is of precisely the same character as B^, 

 but must be regarded as B^ rotated through 90°. Combining Qi with 

 jPi according to operation {v) of §3, we see that for small values of we 

 shall get a network of squares of side /2 = («i + ^i)/ 2 sin {0I2), each 

 square being bounded by the rows of small transparent parallelograms 

 corresponding to B^ and Q^, and having a dark interior. Thus the 

 pattern ttj corresponding to B^ and Q^ combined resembles in its 

 general features the original half-tone plates, i.e., we have dark squares 



