kR 
Radiation and Vibrations in Cortical Pipes. 11 
Applying this to equation (28), we find, after a little trans- 
formation, 
A( cos h\ + h\ sin kr-i) = B {kr x cos br-i— sin h\), 
and A (cos kr 2 + kr 2 sin kr 2 ) = B (kr 2 cos kr 2 — sin kr 2 ) . 
On dividing out by the cosines and writing tan X for kr x 
and tan 6 2 for kr 2 , we may eliminate A/B between the above 
equations. Thus 
. ,-r> tan #i — tan h\ , /r . 7 N //i 7 \ 
A / B =iTt^^ri 1 = tan (^-^ )=tan ^-^>' 
or kr 2 — tan -1 kr 2 = kr l — tan -1 fcr^. . . . (30) 
The transcendental form of this equation shows that the 
nodes are not equidistant in a conical pipe. We will pre- 
sently find where they are in the important case of a complete 
cone with open end. 
Closed Cone. — To treat the case of a cone continued to the 
vertex and with base closed, we have simply to write r ± = 
in equation (30) and R as the slant length of the cone for r 2 . 
This gives 
timkR, = kU (31) 
To solve this equation, which we may regard as tan#=#, 
we may proceed graphically. Thus plot the two graphs y = x 
and y — tan x. Then their intersections will give the roots 
required. See fig. 1, p. 80 as an illustration of this. The 
equation may also be solved by successive approximations by 
which (in another connexion) Lord Rayleigh finds (' Theory 
of Sound/ vol. i. p. 334), 
=tf/7r=0, 1-4303, 2-4590, 3'4709, 4'4747, 5*4818, G-4844, &c. ] 
= Qi, 02, 0z, Q±, 0-o, 06, 0i, say, J 
Thus these quantities, denoted by the 6's, each multiplied 
by 7r, give the first seven values of kJl in equation (31). 
Now since #i = 0, we may write (knR)=7rd n+ i. But we 
also have as the general relation k H = 27rN n /a. Hence, we 
may write for the frequency of the ?ith tone natural to the 
closed cone 
N,= 2V" 41 (33) 
Thus the frequencies are directly proportional to the speed 
of sound, inversely proportional to the slant length of the 
cone and the relation of the various possible tones in the 
series is defined by equation (32) giving the roots of (31). 
(32) 
