PIPES AND RESONATORS 257 



which side we start. Some fairly obvious precautions are 

 necessary in using the method, and it is easily seen that the 

 convergence will be slow if the two curves have nearly the 

 same inclination (in the same or in opposite senses) to the axis 

 of x. Expressed as multiples of TT, the successive approximations 

 obtained in this way to the first root of (7) are* 



1-5, 1-433435, T430444, 1-430304, 1*430297, .... 



The same analysis can obviously be applied to the theory of 

 vibrations in a conical pipe whose generating lines meet in 0. 

 If the tube extend from the origin to r = a, the usual approxi- 

 mate condition (s = 0) to be satisfied at the open end gives 



sin&a = 0, (10) 



the same as for a doubly open pipe of length a ( 62). For the 

 case of a tube extending from r = a to r = b, and open at both 

 ends, we require the complete solution 



r<f> = A cos kr + B sin kr. (11) 



The conditions give 



Acoska + Bsinka = Q, Acoskb + Bsinkb = Q, (12) 

 whence sin k (b a) = 0, (13) 



as in the case of a doubly open pipe of length b a. 



If x be any solution of the general equation (1), it appears 

 on differentiation throughout with respect to x that the equation 

 is also satisfied by </> = d%/3# . We have already had an example 

 of this in the general double source of 73. From (6) we 

 derive in this way the solution 



d /sinferX ( . 



* =G Wx()> (14) 



or, if x = r cos 6, 



ri d/smkr\ - C ,, , 7 \ Q /- ~\ 

 6 = C=- - - cos 6 - (kr cos kr sin kr) cos 6. (lo) 

 or \ T / r 1 



This leads to another series of normal modes of the air con- 



* In calculations of this kind, and for the purposes of mathematical physics 



generally, trigonometrical tables based on the centesimal division of the 



quadrant are most convenient. A four-figure table of this type is included in 



J. Hoiiel's Eecueil des Formules et des Tables Numfriques, 3rd ed., Paris, 1885. 



L. 17 



