83 



When the original angles cf , (f', a and <t' are again introdnced, 

 the parameter equations become: 



iv = ± l^r* cos* {(p — :f') — a* cotci^ (rp — 7') 



7/ —. zt \^r' cos' \{(p—((') -h («—«') J — «' cotcf \(fp- <p') -f {(( -<c')\ 



{VII I] 



For a constant sum of phase, we tind tlie same equations bj 

 changing (p' and n' into — 7' and — a'. 



In both cases, tlie image is the reflexion of the j)art in the tii'St 

 quadrant with regard to the axes. 



Characteristic is tiie function 



which is real for sin (p ^ 



ƒ {(f) = V r* cos* (p — «' cotg* <p. 

 a 



r 



It has an initial value for q:- = hg sin — , a fast reached maxi- 

 mum for sin* (p= — and it becomes for this maximum =r — a. 

 r 



This is in accordance to the fact, that the circumscribed squares 

 of the partially covering unissons have a common square with sides 

 = 2(r — a), in which square the moire-image is inscribed. 

 For the more general case: 



X z=z 7\ cos 2 (p -\- b X =. r ^ cos 2 fp' — b 



y z=z 7*, cos 2 {rp -\- u) -\- a y '=■ '>\ cos 2 (</)' + f<') — « 



we find : 



;i; = ± V i\^ cos* {<p — rp') — b* cotg^ {<p—fp') 



y=± Vt* cos* \{<p - if') + {a - a')\ - a* cotg* \{rp - <p') -f- {a- a')\. 



Construction of the Hyperbola s. 



The construction is similiar to that, used for a liissAJOUS-curve, 

 that is: straight lines are drawn parallel to the scaled axes of an 

 orthogonal system and the points of intersection are joined diagonally. 



Fig. 1 shows a diagram of the funtion 



f{(p) = Vr* cos* <f — a* cotg* fp 

 for a = S, r = 30. tf is given in units of |f = 3|° and so, the 

 unisson has V=:12 ellipses. 



The maximum ordinate is r — a = 22, for <f ± 30° 



(ƒ (30°) =: K483 while 22' = 484). 



The initial value of r/) = 15°, — being = =fc i. 



r 



6 



Proceedings Royal Acad. Amsterdam. Vol. XXV. 



