78 Prof. E. H. Barton on Spherical 
Open Cone. — We now consider the case of a complete cone 
with base open. At the open base, we have, as before for an 
open end, the condition rs = 0. And at the vertex, the origin 
of coordinates, we have from equation (26) that rs = there 
also. So that although one end is open and the other 
closed, we have the apparent anomaly that the same con- 
dition applies to each. Hence if R, is the coordinate of 
the base, L e. R, is the slant length of the cone, we have 
from (29) 
R = m\J2 or N m =ma/2K, . . . (34) 
where N OT is the frequency of the rath natural tone and \ m its 
wave-length, a being the speed of sound. This then shows, 
what has before been remarked upon as strange, that a cone 
open at the base and closed at the vertex gives practically 
the same fundamental and the same full harmonic series of 
other natural tones as are obtainable from a parallel pipe 
of the same length and open at both ends. Of course, when 
the corrections for open ends are taken into account, the 
statement as to pitch and length suffers a slight modification. 
For the double open-ended parallel pipe has two ends needing 
correction and the cone only one ; moreover their diameters 
may differ. But if a cone and an open-ended parallel pipe 
are prepared of slightly different lengths so that their funda- 
mentals are in unison, then their other partials will be in 
accord also. This may easily be verified by pipes of zinc 
tested with a set of tuning-forks forming the harmonic 
series of relative frequencies 1, 2, 3, 4, &c. 
We see from the first form of (34) that the wave-leneth is 
inversely as the order of the tone produced ; hence the 
antinodes are all equidistant. This, however, does not apply 
to the nodes. To determine the positions of the nodes we 
must refer to equations (31) to (34). !Now equation (33) 
gives in terms of 6 the various values of N for a closed 
cone of fixed slant length R. Let us, however, substitute 
the variable r for the constant K and, dropping the subscript 
of N, rewrite this equation as follows : — 
2Nr/a = 1? a , 3 , 4 , 6- 0j 6 , or <9 7 , &c. . . (35) 
We may now regard both N and r as variables which must 
satisfy (35), r being the slant length of a closed cone. Again, 
equation (34) gives the frequencies of the various tones 
natural to the open cone. Let us rewrite it, dropping from 
'N its subscript and writing for m on the right side the series 
of natural numbers which it represents. We thus obtain 
2NR/a=l, 2, 3, 4, 5. 6, or 7. &c. . . . (36) 
