Curvature and Torsion of a Helix on any Cylinder. 55 
more closely show where the structure is irregular. No meaning 
is attached to the absolute size of the rings. 
The figures indicate the valency of each ring, and consequently, 
the outermost figure of each element shows the number of 
electrons associated with it. 3' represents a rare earth ring, 
which, although drawn close to the ring next it, for the sake of 
distinction, produces no distortion in the element following it. 
Group 3 only differs from group 2 by the fact that the rings 
are trivalent instead of divalent. The structure of the chromium 
group is the same as that of the vanadium group, but the penta- 
valent rings in the latter are replaced by hexavalent ones. 
Manganese is the same as vanadium with a heptavalent outer 
ring. 
The only difference between iron, nickel, and cobalt, is in the 
mass of the incomplete dotted ring. The same holds for the 
remaining members of the iron group. 
The three groups following copper have the same structure, but 
their outer rings are di-, tri-, and tetravalent respectively, instead 
of monovalent. ■ 
Similarly, the structure of the oxygen and fluorine groups is 
the same as that of the nitrogen group, but the rings are hexa- 
and heptavalent instead of pentavalent. 
III. On the Curvature and Torsion of a Helix on any 
Cylinder, and on a Surface of Revolution. By L. V. 
Meadowcroft, B.A., M.Sc, Scholar of Trinity College, 
Cambridge *. 
IN the following short paper I propose to discuss the 
curvature and torsion of a helix on any cylinder, and on 
a surface of revolution, by a new method. The method is of 
general application, and will apply to a helix on any surface. 
In this latter case, however, the formulae are somewhat long 
and unwieldy ; and so I have contented myself by simply 
indicating the method of procedure, without giving the 
analytical details in full. As far as I know the results 
are not new, but the method presents several points of 
novelty and interest. 
I define a helix as a curve for which the ratio — is 
constant, p and ar being the radii of curvature and torsion 
of the curve at any point. It follows, by a theorem given 
belowj that the tangent and binomial at any point of the 
curve make fixed angles with a fixed straight line in space. 
An equivalent definition is that a helix is a curve such 
that the tangent at any point of it makes a fixed angle with 
a fixed straight line in space. 
* Communicated by the Author. 
