1897-98.] J. Boeke on Derivation of Curves of Vowel Sounds. 89 
spot ; or, in other words, in derivating from them the curve of the 
sound which produced the impression. 
This derivation does not give strictly accurate results, but the 
faults are too minute seriously to influence the character of the curve. 
Simple reasoning will show that the depth, d , of the impres- 
sion on a certain point may be easily deduced from its breadth, b, 
on the same point. Suppose EGDF[(fig. 1) to be the transverse 
E 
section of the cylindrical marker • HK the longitudinal section of 
the surface of the wax cylinder ; ACBD the transverse section of 
the groove which the marker ploughs in the surface of the cylinder, 
AB — b being its breadth ; CD — d being its depth ; and EG — 
ED = 2 r being the diameter of the marker. 
Obviously, if the axis of the recording marker were tangent to 
the surface of the cylinder, so that the prolongation of its edge 
would cut the axis of the cylinder, we should have the equation : — 
(±bf = d( { lr-d) 
or + = 0 . . . (1) 
from which we deduce : — 
d = r± J(r + ^b)(r-±b) . . . (2) 
As the equation (1) is that of an ellipse, the axes of which are 
