Manchester Memoirs, Vol. Ixiii. (191 9) No. 4 9 



of the line AA^^^. Whether i be positive or negative we shall have 

 f sin^ Q = n'^d^^ + d^^ - 2nd^ ^. cos $ 



= «^^i^ + (« + (.y ^r - 2nd^^ {n + f ) cos 6* • • (■^"0 



For small values of $ we can put cos (^ = i - (sin=6') / 2, and the above 

 expression reduces to 



f- sin^ ^ = «=^r [(I +^)sin^^ + ^] 



Thus for small values of we have 



p sin ^ = £^, [i + ^sin^^ (1+^) J . • (xiv) 



Also 



Hence 



sm^^_nd^ , whilst A = -^ sin ^ cosec 

 sin p sin n 



y..^ - ^^^ .... {xv) 



£" + ^(«^ + c«) sin^^ 



and we see that this has the limiting value 



p^ = ^ {xva) 



£ 



when becomes zero. Thus the pitch /. does not become infinitely 

 ^reat when it reaches its greatest value at d = o\ 



The result for/2 can also be obtained by reference to Fi^. 2a, where 

 A^A = (^2 + i>2) cosec 0, and AiA^^^ = n{a^ + b.,) cosec 0. 



Clearly — p^ . AA^ = area of triangle AA^A^^^ 



2 



= ^n {a^ + b^) {a^ + b.) cosec Q. 



^, . (a, + b,) {a^ + b^) 



Thus /o = -^— ^ ^4-^: 



p sin d 



Jri^ + (n + lY - 2n{n + e) cos $ 



, exactly . (xvi) 



From this equation, the equations (xv) and (xva) can be easily 

 deduced. 



The results of this section are easily exemplified by taking two 

 intaglio screens of nearly equal pitch, and superposing them at small 

 angles. It will be found that ^ = o gives a finite pitch corresponding 

 to equation (xva). It will also be found that as passes through zero 

 from positive small values to negative small values, the orientation of 



