Vol.. 8, 1922 
PHYSICS: A. G. WEBSTER 
103 
V 
V 
e 
o 
we have 
dx = QOS'ir/2i}'^dv, dy = smTr/^v'^dv, ds^ = dx"^ + dy"^ = dv^, 
dy/dx = tan^ = t^mr/^v^, 0 — -wj^fo'^ = -kI^s^ = as^l2. 
The integrals above have been tabulated by Gilbert, and are reproduced 
in Verdet's Tratte d'Optique Physique. From them the spiral has been 
plotted, and is shown in the outer curve in figure 1. It will be seen that it 
differs very little from the elastica. As a matter of fact the ratio of 
the maximum ordinate at right angles to the chord is .2861 compared 
with .2757 for the elastica. For the circle this ratio would be only .2071. 
The reason for the near agreement of the tw^o curves, in one of which the 
curvature is proportional to the arc, and in the other to the ordinate, 
is of course the fact that for a considerable distance the curve is nearly 
straight. 
The simplest assumption that could have been made for a rough cal- 
culation would have been that of a parabola, y — ax"^, but the ordinary 
parabola of order two would not do since its curvature does not vanish 
at the origin, and diminishes as we leave the origin. A cubical parabola 
would be the most obvious, but we shall do better by putting y = ax^ 
and determining the exponent n so that the curve shall pass through the 
vertex of the elastica and be tangent to it. We find n = 3.458. The curve is 
drawn dotted on figure 1, and lies between the other two curves. The ratio 
of the extreme distance from the chord to the length of the chord in the 
case 7i = 3 would have been .25. We consequently see that any of the 
four curves, the elastica, spiral of Cornu, parabola of order 3.45S, 
or cubical parabola, will give a very good approximation to the actual path. 
We have now to consider the effect of the finite length of the wheel- 
base, and this leads us to consider the question of steering in general. 
We shall first suppose for generality that both the front and rear wheels 
can be steered, as in a hook and ladder truck, but we shall still neglect 
the effect of the finite width of the track. The case contemplated is ex- 
actly realized in the bicycle. In figure 2, let I be the length of the wheel- 
base, let s, 6, a, p be the distance run, the angle made by the tangent with 
a fixed direction, the angle through which the wheel has been turned (that 
is, the angle made by the tangent to the path with the chassis, and the 
radius of curvature of the path, respectively, for the front wheel, and let 
the same letters wdth accents denote the corresponding quantities for the 
rear wheel. (When we come to consider the dynamics of the matter we 
shall need corresponding quantities for the center of mass, and shall de- 
note them by two accents.) Then since the whole machine turns about 
the instantaneous center which is the intersection of the axes of the two 
