Mathematics. — “Explanation of some Interference Curves of 
Uniavial and Biaxial Crystals by Superposition of elliptic 
Pencils”. By ‘J. W. N. re Heux. (Communicated by Prof. 
Hk. pr Vrits). 
(Communicated at the meeting of February 20, 1921). 
In Lissasous’ “Etude optique des mouvements vibratoires’*), the 
name of ‘Unisson” is given to the curve, resulting from the com- 
position of two vibrations, which only differ in amplitude and in 
phase. 
When the amplitudes are supposed to be equal and not diminishing 
with continued movement, when the directions are at right angles 
and the difference of phase increases from 0° to 90° — the unisson 
may be considered asa pencil of ellipses, whose envelope is a square ’). 
The null-ellipse of this pencil is a diagonal d, of the square, the 
end-ellipse is the circle, inscribed in the square. Let the other diagonal 
be d,.’ 
Two equal unissons, U, and U,, partially covering each other, 
produce certain ‘watered curves” (moiré), which may be divided 
into two sets: 
1°. those similiar to hyperbolas, when the exact covering of U, 
and U, may be obtained by moving the centre of the pencil along 
d, (fig. 1) and | 
2°. those, similiar to lemniscates, when the exact covering may 
be obtained by moving the centre along d, (fig. 2). 
The “watered curves” *), above mentioned, bear a strong resemblance 
to the interference curves of some crystals — it will be examined, 
whether these interference curves may be explained by superposition 
of two pencils of ellipses. 
Theretore, the image of the hyperbolas will be compared to the 
interference curves of a uniaxial crystal in convergent light, the 
erystal-plate being cut parallel to the optic axis and the image of 
the lemniscates to the interference curves of a biaxial crystal, the 
plate being cut perpendicular to the first diameter. 
1) Annales de Chimie et de Physique, 3i@me série. t. Ll. Octobre 1857. 
2) Proc. Kon. Acad. v. Wet. pp. 857—870 March i914. 
Mathésis, 3i¢me série t. X pp. 209—212, 1910. 
3) On stereoscopic curves, see Comptes Rendus t. 130 p. 1616. 
Also: Harmonic Vibrations and Vibration Figures by J. Gooup, C. E. BENHAM, 
R. Kerr and Prof. L. R. WiLBERFORCE, Newton, London. 
