












Marcu 31, 1923] 
and the length of the chain to be undistinguishable, so 
that the tension 7'=ox, as at rest, o the line density. 
Putting g = w*/, where / is the height of the equivalent 
conical pendulum revolving at the same rate w, the 
uation of relative equilibrium is 
having a solution y=4F(x/l), where F(x) is Fourier’s 
ction, defined by the series 
— x) 
(1k)* 
The sort called Furniture Chain is suitable for ex- 
ent: the links are small hollow brass spheres, 
joined up by rivet links, and it is sold in various sizes. 
A length of the large size of about 4 feet is suitable 
for whirling round by hand, and producing a curve in 
[, 2, 3, 4, . . . waves, showing to the eye the position 
f the first roots of F(x)=o. 
_ Standing up on a chair or the table, a length of 8 or 
F(x) = vo 
the dynamobile toy. The chain springs at once into a 
series of waves, where the higher roots are seen and 
their spacing, prolongation of the figure at the end of 
Gray. 
The chain can also be used to show off a real catenary 
curve, instead of the string recommended in Routh— 
much too kinky and destitute of flexibility to form a 
good catenary. And dropping the chain from a height 
is a good problem on a steady blow, equivalent of a series 
of impacts of the discrete links. 
With a rotating chain of variable density, ox”, the 
tension T=ox"*1/(n +1), and the equation changes to 
r+ + fox) dx=o, 2+ (n+ + (n+ 179 =0, 
the solution of which may be written 
y=0( - 5) Fa + 175 
and /is the length of the subtangent at the lowest point. 
For if «= F(x) is the solution of the equation 
Pu du, 
Pe + aR +u=o, 
differentiating m times, with y=(-d/dx)"u=F,,(x), 
2+ (n+) +y= =0, y=F,(x)= See 
the Fourier function of the nth order. 
tion into an integration, making F_,,(x)=x"F,,(x). 
F,(-*). 
NO. 2787, VOL. 111] 
12 feet of the smaller size may be set in rotation by. 
Here m may be changed into — 1, and the differentia- 
Gray’s function I,(¢) (p. 20), is the equivalent of 
NATURE 
i 
423 
But with the variable s=2 ,/x of the Bessel form, 
the equation changes to 
209+ (n+ ys + 2ty= 0, 
and this, with y=(42)~ "z, into 
ne + au + (22 —1®)u=o, 
defining u=J,(z) =(4s)"F,,(42”), and the simplicity dis- 
appears of the derivation of J,(s) from J,(z) by suc- 
cessive differentiation or integration ; factors intervene 
of powers of z. 
The interlacing of the roots of F, and F,,,, is evident 
from the differentiation ; and there is an infinite series 
of positive roots, but none are negative. 
This chain of variable density could be imitated by 
a flexible lattice blind, of appropriate curvilinear out- 
line, hanging vertically, and rotating bodily. 
Lecornu’s problem of the oscillation of a large 
weight, raised or lowered by a chain of which the 
density may be neglected, is seen in operation in the 
erection of the tall buildings springing up around ; it 
gives rise to similar expressions. 
A Fourier function of fractional order arises in the 
question of the stability of a tall mast or tree, or of a 
chimney stalk when it begins to flinch on the foundation, 
and starts to curl over from the vertical ; illustrated 
experimentally by a thin steel wire clamped in a vice. 
With uniform cross-section, the equation is 
ee T® 5 xp oO, 3 oat aa °, 
where p=<dy/dx, k is the radius of gyration of the hori- 
zontal section across the plane of flexure, and e is 
Young’s elastic length of the material, quotient of the 
modulus of elasticity divided by the density. 
Every linear differential equation of the second order 
is reducible, by a change of independent and dependent 
variable, to the canonical form 
and when the differential invariant I=kx", any power 
of x, the form to which Riccati’s equation is reducible, 
the equation is reduced to Fourier’s form by a mere 
change of the independent variable to 
Rkym+2 
= nea 
and becomes Fourier’s equation for 
I 
u=F,(3), m+2 

a= -— 
(Watson, p. 88). 
Here with the uniform column on the verge of droop- 
ing from the vertical, m=1, p=bF _(x3/gek?). 
