424 
NATURE 
[Marcu 31, 1923 


The smallest root of F_,=o is about 0-88, say § 
(Watson) ; this makes the critical height 
x= 2(eh®)) = (hed”)s 
for a circular rod of diameter d. 
For steel, we may take e=250 million cm, one 
quarter of a quadrant Q of the Earth, 3/$e=500. 
With a steel wire held in the vice vertical, one milli- 
metre in diameter, d=o-r cm, the critical height 
x=500 d'=107-7 cm, a little over one metre. As the 
height is increased through this length, the vibration 
becomes sluggish more and more, and finally ceases, 
and the wire droops. 
The drooping of a candle on a hot day will give an 
illustration ; also a field of corn when it is ripe, where, 
to obtain a complete solution, the weight of the head 
would require the introduction of the Fourier function 
of the second kind, or a Bessel-Neumann-Weber function 
(Gray, p. 14; Watson, p. 308) ; so too for the addition 
of a weight at the end of the vibrating chain. 
Here the flexural elasticity keeps the rod, mast, 
or cornstalk vertical ; a flexible chain cannot be made 
to stand upright ; the sign of x would be changed in the 
relation, and the Fourier function has no negative root. 
But a quasi-rigidity can be imparted, as in the re- 
ported rope trick of the Indian juggler magician, if our 
chain carries a gyroscopic flywheel in rapid rotation 
inside each link, like a pile of spinning tops, and then, as 
shown in Phil. Mag., Nov. 1919, p. 506, the differential 
equation of the former result changes to the form 
(pat T=? 
with « measured downward from the free end at the 
top; the solution is 
and the first value of y=o is given by (a—«)/l=1-44. 
Thus a length « of the gyroscopic chain can be made to 
stand upright, given by x=a—1-44 l. 
The whip and whirl of a revolving shaft has become 
a question of practical importance in the swift-running 
machinery of a turbine, internal-combustion flying- 
machine motor, and gyro-compass. 
Here it is obvious that the shaft will depart from 
the straight form when the revolutions are equal to the 
lateral vibrations of the shaft at rest, held between 
the same bearings, the disturbing and restoring force 
being the same in the two cases. 
The more general form of the differential equation, 
required when the cross-section and density varies as 
some power of x, will be 
£ (x) +kxmy=o, 
NO. 2787, VOL. 111] 



and hence a change to the independent variable 
hxm-at2 
° (m— q+ 2)" 
will lead to Fourier’s equation of order 
ee: 
m—g+2 
The general solution of Riccati’s equation is thus 
expressed by the Fourier function ; and the condition 
that Riccati should have a solution in finite terms 
requires 7 to be half an odd integer. 
Beginning with x= — 3, 

(—x)* 
F_(x)= ai 7( — $) TR’ 
Fob tp 
and a phase angle « may be added to the variable 2 /x 
to include both forms of the function. 
Then the other Fourier functions of order half an odd 
integer are derived by an integration or differentiation 
with respect to x: 
JTF) = - 
EJP), 
JF x)= - wns sine Ft oF _,(2), 

and so on. 
The same simplicity of derivation is not so obvious 
in the table (Gray, p. 17) for J, ,(z), although the sines 
and cosines are replaced by sin, cos (g+€). 
Functions of this fractional order are of frequent 
occurrence in mathematical physics, as in the vibration 
of a sphere (Love’s “Elasticity,” Lamb’s “ Hydro- 
dynamics ”’) for the functions y, and W,, solution of 
(v?+m)$=0, in the propagation of spherical waves 
or the conduction of heat; also for the function Fy 
of Bromwich and yp of Macdonald in electromagnetic 
waves; simplicity would be obtained if all these 
functions were referred to the Fourier form and classed 
there (Phil. Mag., Nov. 1919, pp- 508, 526). 
The Fourier function comes in useful for the dis- 
cussion of a long flat tidal wave in an estuary or channel 
of vertical cross-section K, and surface breadth 5, 
treated as slowly variable, on the assumption of K 
and b varying as x? and x™, simple powers of ». 
The Fourier function is suitable, too, in the discussion 
of diffraction (Gray, Chapter XIV.), provided the area 
of a circular fringe is taken in the formula instead of the 
circumference or diameter. 
The derivation, by differentiation and integration, of 
the Fourier function of different order marks it out as 
more appropriate than Bessel for the passage, in Lord 
Rayleigh’s manner, of the tesseral harmonic P,,#() 
direct into a Fourier function F,(7= 
}m*r) as the order 
oe ae 

