280 PHYSICS: A, G. WEBSTER 
Conical tube, a = aox^ 
72 
l4 + ?^ + ^^^ = 0 (29) 
dx X dx 
cos,kx ^mkx , /sin^x , cos , cos sin ^jc 
u = , u = u = — \ + J, u = — — 
R X 
r, _ 1 D - ^ D - 
■^1 — 72 2' —72 2' -^2 — 72 > 
_^ _ COS kl sin n _ 
— 72 + 71 2 » "^5 — 72 
k X1X2 k X1X2 k XiX2 
and if we introduce two lengths €1, €2, defined by the equations 
tan kei = kxi, tan kei = kx2, 
we easily get 
^^^ sin^(; + .,) ^ j^^^isin*/, (30) 
0:2 sin <Ti X2 
0-2 ^1 sin ^ (/ + €1 — €2) , U2 xi sin k {l — ... . 
c = — , a = , (31) 
j8 X2 sin kei sin ^€2 o-i 0:2 sin ke2 
„ sin ^ (/ 4- €1) , i3 . , . 
Zi ^ + — sin kl, 
„ _ /3 sin kei 
Z2 — — 
0-2 „ sin ^ (/ + 61 — 62) /3 sin k (l — €2) 
Zi ; ; + — ; 
sin kei sin ke2 o-j sin ke2 
„ sin k {I — €1) . ^ . ,j 
Z2 ^ — + - sin kl 
^ ^ _ sin ke2 (T2 (32) 
cTi sin k{l -\- ei —€2) , /3 sin ^ (/ + €1) 
^2 — : : n ■ 
sin kei sin k€2 <J2 sin ^€2 
The formulae (31), (32) were used by Professor G. W. Stewart in designing 
horns to be used during the war. 
It is not true, as is frequently stated in books on musical instruments, that 
the brass instruments of the orchestra are hyperbolic in profile, but I have 
found for all practical purposes the bell of every instrument may be repre- 
sented by one of the three formulae 
Even if an equation cannot be given to the profile the differential equation 
may be easily integrated graphically, or the length may be divided up into 
sections and different values of n used for different sections, as is customary 
in the theory of ballistics. 
