1897-98.] J. Boeke on Derivation of Curves of Vowel Sounds. 95 
while making the measurements through the microscope, by each 
shock through of a circumference, was chosen because the 
periods of my own voice, while speaking, required about forty shocks 
of drum, Q, or -j~ of the circumference of the cylinder, arid because 
forty ordinates were required to make the Fourierian analysis of the 
curves by means of Prof. Hermann’s “ Schablonen.” 
The numbers of the measured breadths were dictated to an 
assistant and written at first upon loose papers, but afterwards in a 
book, which proved more satisfactory, as it was sometimes difficult 
to follow them afterwards in the vastly increasing number of loose 
papers. Of course, it was exceptional when the periods required 
exactly 40 shocks of drum, Q, and as the number of shocks varied 
between 20 and 60, it was necessary to divide the curve obtained 
into 40 equal parts, so as to be able to submit its 40 ordinates to 
the harmonic analysis. 
As it took a good deal of calculation to obtain this result, all 
periods were drawn in a constant length of 200 mm., and once 
for all, a number of strips of millimetre-paper were made, each of 
them measuring 200 mm. in length, but divided into a different 
number of equal parts, varying from 20 to 60 parts, with the 
number inscribed on them. These strips were sufficient to draw 
the whole curves of any pitch in a length of 200 mm. 
From the measured breadths of the period were deduced the 
depths corresponding to them, by means of the above-mentioned 
list, and, by multiplying the numbers obtained by a suitable factor, 
all the ordinates were reduced to whole numbers not exceeding 
20 to 30. The ordinates were marked on the millimetre-paper 
by means of dots in their proper places, by using the strips referred 
to, and then the curve was drawn with a finely-pointed pencil 
through the dots as smoothly as possible. When the curve was 
thus obtained, the length of its ordinates, 5 millimetres apart from 
each other, was determined by means of an eyeglass; and the 
forty ordinates so obtained were inscribed on a sheet of centimeter 
paper as described by Prof. Hermann.* It was found most con- 
venient to mark the ordinates themselves with red ink and the 
values of their cosines with black ink, in which way mistakes 
are easily prevented. 
* P finger's Archiv., 47, p. 51, 
