106 Proceedings of Royal Society of Edinburgh. [sess. 
If this method is correct, it will practically deliver us from 
the consideration of another feature of the phonograms of vowels 
which would otherwise now demand attention. This is the normal 
tendency to a decline in amplitude, which was exhibited in fig. 2. 
The centre-of-gravity calculation assumes, in its two simpler forms, 
that normally, and apart from resonance, the amplitudes of succes- 
sive partials would be equal ; but we have seen that the series a', 
a'\ a ", etc., is normally a rapidly declining series ; and in any 
calculation based on them, some correction should be made for 
this. But when our calculation is based upon the series a'p\ a'p", 
a"p"\ etc. ( i.e.j upon amplitude multiplied by frequency), there is 
no such tendency to rapid decline, and our calculation is very much 
freer from this risk of error. Compare the a series with the ap 
series in fig. 2, thus : — 
1 234 56789 10 
a series, . . 38‘0 20*6 19T 7*7 5*4 3*6 27 1*6 0*5 0*8 
ap ,, . . 38-0 41*2 573 30'8 27*0 21’6 18'9 128 4*5 8*0 
Comparing the three methods discussed, it is easy to see that 
Pipping’s method will always give a result somewhat lower than 
Hermann’s; and that the method just recommended will give a 
result somewhat higher than Hermann’s. In the present case 
(fig. 5) the difference between the three calculations is not great. 
The first has given 737 v.d. ; the second will be found to give 
732 v.d. ; and the third 741 v.d. And when the partials involved 
are still higher than these, the differences become more and more 
inconsiderable. But when the lowest partials are involved, the 
differences are large enough to be of vital importance. Another 
difficulty, which I have found in practice, in dealing with rein- 
forcements of the lowest partials, is that when one of the reinforced 
partials is the fundamental, the results drawn from different 
phonograms of the same vowel seldom closely agree ; and I have 
been led to conclude that the fundamental, though it may be 
either damped or reinforced by resonance, is much less absolutely 
under the control of the resonances, and therefore much less trust- 
worthy as a criterion of resonance, than any other partial. There 
are good physiological reasons to account for this : for the genesis 
and period of the fundamental is definitely associated with the 
