1225 
Now, it is possible by means of a simple algebraic transformation, 
to derive these concentric pencils E,, and E,, from the excentric pencils 
Ug sand .0-. 
The curves of the new pencils however are composed of four 
vibrations, which differ only in phase and whose directions two by 
two are parallel to those of the compounding vibrations of the unissons. 
To support these conclusions by experiment, | have made use of 
an instrument') planned by myself, with four pendulems, two of 
which describe a Iissasous-curve on a plane, which deseribes such 
a curve itself. In our case, those two curves are unissons. The resulting 
movement is a spiral, which may be considered as a pencil of concentric 
circles. By altering the difference of phase, the circles are converted 
into ellipses. 
It is shown now by experiment, that always hyperbolas are 
obtained by superposition of two pencils, which have for null-curves 
a circle and an ellipse, having the same tangent in the extremilies 
of the minor axis, and lemniscates, when the null-curves are two 
ellipses, one of which is surrounded by the other. 
Because the pencils U, and U,, by inversion, may be derived from 
the pencils B, and E,, it is evident, that the ‘watered curves” of 
the unissons are approximate images of interference curves of uniaxial 
and biaxial crystals. 
This theory is supported by further experiments. The centre of the 
hyperbolas is displaced by a small rotation of one of the unissons. 
The image resembles the interference-curves of a uniaxial erystal, cut 
non-parallel to the optic axis (fig. 3). 
With a small number of curves, the image of the lemniscates has 
both poles surrounded by the inner curve, just as is taught in erystal- 
opties. (fig. 4). Two concentric pencils of circles (each compounded » 
of four vibrations), show the phaenomenon of Newton’s rings. 
Probably it is also possible to obtain Airy’s spirals in this manner, 
and the way, in which a certain image appears by superposition of 
two pencils of ellipses may throw more light upon the phaenomena 
of refraction and polarisation of light in crystals. 
1) Other instruments are described in: Harmonic Vibrations and Vibration Figures 
by Goorp etc. 
Also: A. C. BANFIeLD. The Photo-Ratiograph, Illustrated London News, 
Sept. 25th 1920. pg. 470. | 
Comptes Rendus t. 130. pg. 1616. 
H. J. Oosting. Hand. XIV Ned. Nat. en Geneesk. Congres March 1918. Ann. 
d. Phys. u. Chem. N. F. 33. p. 415 1888; Maandbl. voor Natuurwetenschappen 
1898; Zt. f.-d. Physik. u. Chem. Unterr. 8 p. 190 1895 and 11, p. 221 1898. 
73 
Proceedings Royal Acad. Amsterdam. Vol. XXIII. 
