Manchester Memoirs^ Vol. Ixiii. (1919) No. 4 



{iv) Repeat operations using sets Y-^ and X^ 

 pattern be denoted by Q^. 



Let the resulting 



{v) Let small pieces of white paper be cut out exactly corresponding 

 to the transparent spaces of P^., Q^, F^, and Q.. Let these be arranged 

 exactly as in their corresponding patterns ?„ Qi, P^^ Q^, and in their 

 proper positions and superposed (i.e., pasted over one another) on a 

 black background. The final effect will be pattern P, if it is under- 

 stood that white patches correspond to transparent spaces in P. 



Fig: I. 



To prove the truth of the above constructions, we observe that (a) 

 the whole of the transparent spaces of P„ Q^, P^, and Q^ must form 

 part of P, whilst (/') the whole of the transparent spaces of P are 

 included in those of P^, Q^, P2, and Q^. 



We have thus analysed the composite pattern P into /our simpler 

 patterns, viz., the results of operations (/) to (tv). 



^4. We proceed to discuss the pattern P^, which is the result of 

 superposing sets Xj^ and X^ of transparent bands, at some angle d (say). 

 In P/g. I is shown the effect of superposing two sets of bands and it is 

 seen that the pattern consists of a series of transparent parallelograms 



