750 PROFESSOR FLEEMING JENKIN AND J. A. EWING ON THE 
after transcription served also as a continual check on the fidelity with which 
the phonograph itself was registering the spoken sounds. | 
After the curves had been drawn they were subjected to harmonic analysis 
to determine the amplitudes of their constituent partial tones. The curves 
may be regarded as giving a graphic representation of a functional relation 
between 2 and y where these are rectangular co-ordinates, the axis of x being 
parallel to the line joining successive maxima or minima. This function, being 
periodic, may by Fourrer’s theorem be represented by the well-known 
expression— 
y=A,+ Zz, a sin (nz + B), 
where A, is a constant depending on the position chosen for the axis of 2 and 
the terms under the sign of summation are simple harmonic constituents cor- 
responding to the partial tones of HELmMHoLTz: the prime being given when 
n=1, the second partial or octave of the prime when n=2, the third partial a 
twelfth of the prime when n=3, and so on. The successive values of a are the 
amplitudes of the several partials, and 8 their phase. The above expression 
may be written— 
y=A,+A,sin z+ A, sin 27...... +A, SIM m@+...... 
+B, cos #7+B, cos2z7...... +B, c0s 20+. 1. 
where A and B are such that 
oe SSSR ee -1 
a,= AZ +B? and B,=tan ;*. 
By drawing and measuring a number of values of y for a given periodic 
curve, a corresponding number of values of A and B can be evaluated. Our 
process of analysis was as follows :—A straight line was drawn as axis tangent 
to two successive minimums in the transcribed curve, so as to include one 
whole period. The length of the period was determined by comparison of 
several on each side of the one drawn, and a portion of the straight line equal 
to the period was set out and divided into twelve equal parts. From these 
divisions perpendicular lines were drawn cutting the curve. The length of 
these lines between the axis and the curve were measured in two-hundredths of 
an inch, by means of a scale graduated to twentieths (the decimals being esti- 
mated). The numbers obtained in this way formed twelve values of y for the 
one period chosen, and afforded the data for calculating the amplitude and 
phases of the first six constituent partial tones. Professor Tarr was kind 
enough to supply us with the solutions of the simultaneous equations for twelve 



