108 Proceedings of Royal Society of Edinburgh. [sess. 
show that these two reinforcements are due to two separate reson- 
ances. The more natural thing, in the absence of other evidence, 
is to see in these two strong partials the evidence of a strong 
resonance, about half way between them (1050 v.d.), reinforcing 
both. Every phonographic investigator has been deceived more or 
less in this way. What we really want, in such a case, is to have 
a greater number of partials within the sphere of the resonances, 
' to show us more clearly where each resonance culminates ; and the 
only way to have more partials is to sing the vowels at a lower 
pitch. In the instance just given there is no probability of seeing 
any sign of the doubleness of the resonance until we sing the 
vowel as low as 150 v.d. Then the resonance of 900 v.d. neces- 
sarily culminates on the 6th partial, and that of 1200 v.d. on the 
8tli, and there is the 7th partial between, to show, by its want of 
reinforcement, that we have before us the effect of two reinforce- 
ments, and not of one. But in practice this is no more than a 
probability, for it may, and does, happen that the middle partial 
gets reinforced from both sides, and thus comes out with a higher 
strength than that of either of the real culminations. It will be 
best to corroborate the principles just started by actual example, 
and then proceed to consider this problem of the intermediate or 
common partial in a more general form. But enough has been 
already said to show that this problem becomes less and less 
troublesome the lower the vowel is sung. If, for example, in the 
instance last supposed, the vowel was sung at 100 v.d., there would 
not be one intermediate partial, but two ; and it will be seen shortly 
that the chance of both of these receiving concurrent reinforcement 
from both resonances is slight. The resonance culminating on the 
9th partial (900 v.d.) would in most cases be distinctly marked off 
in the analysis from that culminating on the 12th partial (1200 
v.d.), by the marked weakness of the 10th or 11th partial, or both. 
For actual example let us first compare with fig. 5 the analysis 
of the same vowel, sung at the same time by the same person, at 
the much higher pitch of 247 v.d. (fig. 7). For the sake of com- 
parison the diagrams will continue to be set out on the same 
principles as hitherto, though in strictness the intervals ought to be 
set out as in fig. 6, and the perpendiculars ought to represent, not 
simple amplitude (a), but amplitude x frequency (ay). 
