Mathematics. — ''Explanation of some Interference -Curves of 

 Uni-axial and Bl-axial Crystals by Su})er position of Elliptic 

 Pencils." (Tliird paper.) By J. W. N. Le Heux. (Cominiiiiioated 

 by Prof. Hendutk de Vries.) 



(Communicated at the meeting of March 25, 1922). 



Some well-known inleiferenee-cuives, f.i. the liyperbola's and the 

 leiiiniscates are obtained by superposition of two equal unissons, 

 under certain conditions, as was remarked in my first paper ^). 



From this observation we may derive a parameter-equation for 

 both cases, which enables us to construct the curves in a simple 

 manner. 



The axes bein^ at right angles, the unisson may be given by 



X -=2 r cos 2 (p 

 y =z r cos 2 {fp -|- a). 



Each value of the phase-differerice 2rf correspoiuls to an ellipse; 

 when we suppose, that this phase-difference increases each time 



rr 



with 2(c = — , the unisson has n ellipses. 

 2n 



With regard to an easy construction, the angle 2'/ may also be 



supposed to increase with — . 



Zn 



The two equal unissons, partially covering each other, are given by : 



.V = r cos 2i ff -\- a | 



y = r cos 2 {rp j- «)-[-« ) 

 .f =z r cos 2 (p — a i 



y :=z r cos 2 {rp' + <() ~' ^\' 



where a is constant and <[ r. 



The distance between the centres is 2a\/2. 



By altering 2a (and also 2ii') from to , the image of the hy- 



perbola s is obtained, and from — to .tt, that of the lemniscates. 



JL 



{I) 

 {11) 



1) These Proceedings Vol. XXIII, p. 1223—1225. 



