85 



some experiments, while another time, under apparently the same 

 conditions, a family of ovals appeared. At last, it was fonnd, tfiat 

 tlie angle between the planes of the pendulums cansed the difference : 

 the image of the lemniscates is not bnilt, nnless this angle differs 

 from 90° and with a very large nnmber of ellipses per nnisson. 

 So in lig. 6, where the angle between the directions of the two 

 composing movements is ± 145° and each nnisson has rfc 120 ellipses. 



The reason for this large nnmber of ellipses proceeds from the 

 swift rising of the fnnction in fig. 1. Dividing-points near the centre 

 are not obtained, nnless the interval 4 — 5 is divided into f. i. 15 

 parts, corresponding to a phase-difference of -4°= 15' and a nnmber 

 of 180 ellipses per nnisson. 



The experimental nnmber however is limited in conseqnence of 

 the thickness of the ink-lines. 



The phaenomenon is mathematically explained as follows: 



The nnisson 



x" = r cos Icp. 



y" = r cos 2{(p -\- a) 



upon a system with angle 2pr, becomes npon an orthogonal system 

 with the same bisectrix : 



.^•' = r sin (/? + 45°) cos 2 (f -\- r cos (/? + 45°) cos 2 {(p -f «) ) 



y' — r cos {i3 -f 45°) cos 2 ff + r sw (p? + 45°) cos 2 (rp + a) \ ^^^^ 



When ^ -j- 45° = 7 and seeking the equations of the moire-image 

 ill a similiar manner as before, the composing (oblique) imissons are : 



X =z r sin y cos 2 (p -\- r cos y cos 2 {(p -\- a) ^ a 

 y :=r cos y cos 2 (p -\^ r sin y cos 2 {(p -|- a) -\- a 

 and 



.1; = r sin y cos 2 (f> -(- r cos y cos 2 (7/ -f «') — a 

 y :z^ r cos y cos 2 r// -|- ^ siji y cos 2 (</)' -)- «') — a 



and a point of the moire-image has the parameter-equations: 



X sin y — y cos y 



cos y — sin y 

 X cos y — y sin y 

 cos y — sifi y 



= zt [/r' (I + sin 2 y) cos' {<p — f/') — a^ cotg^ (<p—q') 



" =t I/,,' (1 ^ sin 2 y) cos' \ (<f—cp') + (« — «') | — a' cotg' \ {fp—(/ ') + («—«') | 

 Now 



=b Vr^' cos'' {<p—(p') — a- cotg* {<f—<p') 



y — zt y^r,' cos' \(<p--<p') + ia — (t)\ — a' cotg' \ (<p — <fi') + («- a')\ 



where 



6* 



