Velocity of Sound in Air. 451 
length of pipe AB to speak a note of a certain pitch, say 105 
vibrations, and let BO be the half wave-length of pipe added 
to AB ; the three-quarter wave-length AC being tuned to 
I 
speak the same note as AB. Assuming values in four typical 
cases, we obtain the following results (p. 452), in working out 
which arithmetical means instead of mean proportionals have 
been taken for the sake of numerical simplicity, although per- 
haps the latter would be more correct. 
In Cases 1 and 2 the length NN 7 is the true length of a 
half-wave of 105 vibrations. In (1) the position of N (the 
node) would remain unchanged on the addition of the half- 
wave ; but in (2) the three-quarter wave would be of 106f , 
and N would change its position to W: this, however, would 
introduce no error in the result, for 3(106§) — 110 = 2 x 105, 
the length for half-wave as determined by the positions of 
N and W. 
In these two cases it is assumed that the first quarter-wave 
has equal constraining-power with the added quarters ; but 
this is not strictly correct for cylindrical organ-pipes, in which 
the first quarter-wave is shortened and the mass in vibration 
reduced, through the contraction of the opening at the speak- 
ing end. The tendency of this would be to influence the 
results as shown by Case 3, where the first quarter-wave is 
assumed to have a mass of § instead of unity, and in which 
the measured length would be represented by 3(107^) — 115 
= 207i, whereas it should be 2 x 105 = 210. 
In Case 4 (a), suppose the point N to have been determined 
for the pitch as heard, 105, with a quarter wave-length of tube. 
A half-wave is added, and tuned to the original pitch, = 105, 
and "NW is taken as its measure of length. This is correct if 
the relative strength of the partials remains the same ; but if 
a modification such as (5) takes place, the third partial falling 
to half-strength, the position of N as observed would no longer 
be correct for a pitch of 105 but only for a pitch of 103^. A 
further complication would arise in this case, through the 
introduction of the point considered in Case 3. 
To eliminate these sources of error, one of two things must 
be attained : we must either deal with pure tones or be careful 
to have resonating pipes of such form as to have their various 
proper tones strictly in accord with the harmonic series, and 
not merely approximately as is the case with organ-pipes; 
2 12 
