Vol. 8, 1922 
PHYSICS: A. G. WEBSTER 
101 
This does not, however, determine the shape of the curve, to obtain which 
some assumption must be made. If we assume that the mean vahie of 
the centrifugal force or, what is the same thing, of the curvature (since v 
is constant) is to be a minimum we shall get no result, except a straight 
Hne, which would be a discontinuous solution, and inadmissible. We shall 
therefore assume that the mean square of the curvature is a minimum, 
that is 
dfK''ds = bf{de/dsYds = 0, 
subject to the condition 
dx = S cosdds = constant, the length of chord. 
But this gives 
d^d/ds'- + Xsin^ = 0, 
where X is some constant. But this is the equation of the elastica or 
curve assumed by a bent flexible rod, ruler, or spline. In fact, in order to 
obtain the equation of the elastica, we have to make the potential energy 
of bending a minimum, and as this, at each point, is proportional to the 
square of the curvature, the problem is mathematically the same. 
If we take the chord for the X-axis, we shall find the equation satis- 
fied if we take the curvature proportional to y, and introducing the elliptic 
functions of Jacobi, with c a constant of homogeneity, 
dx/ds = cosd, dy/ds = sinS, 
K = dejds = -y/c'',dK/ds = d^d/ds'' = - 1/My/ds = - l/c^sinO,, 
y = acris/c = acnu, dy/ds = — a/csnudnu = ~ 2/?smtdnw, 
where 
s/c — II, a/c = 2k, 
from which 
dx/ds = V 1 _ 4.k^snhidri-u = 2&n'u - 1, 
s/c s/c 
X = 2cfdn'udu - cfdu = c[2E{u)-ti] = c[2E{k,cp) -F{k,<p)l 
0 0 
where F and E are Legendre's elliptic integrals of the first and second kind, 
respectively. The value of k is determined by the fact that at the point 
of inflexion, where = —45°, we have u = K, the complete elliptic inte- 
gral, and since snK = 1 and dnK = ^Jl—h^ we have 2k^IY—k2 = 
sin 45°. Putting k = sina we see that a = 22°.5. With this value for a 
taking values of (p at intervals of 10° from 0° to 90°, values of x and y 
are calculated by means of Legendre's table and the curve drawn, which 
is shown on the inside of figure 1. It fits very accurately a spHne made of 
celluloid, held fast at the vertex, and merely by contact at the point 
of inflexion. W^e find the ratio of the maximum ordinate to the whole 
chord or base is .2757 and that the radius of maximum curvature is the 
chord divided by 2.1429. 
