104 PHYSICS: A. G. WEBSTER Proc. N. A. S. 
wheels, we see on reference to figure 2, \l/ being the angle made by the chassis 
with the fixed direction, 
pdd = (is, p'de' = ds', e a = e' -\- a' = yp. 
We also have 
cosa/p = coscc'/p' = sin (a — a')//, 
which equations give p and p' as functions of a, a'. If then a and a' 
are given as functions, say of 5 and s' , this makes seven equations between 
0 
FIG. 2 
the variables a, a', d, B\ s, s\ p, p', so that we may find the difi'erential 
equations of the path of either the front or rear wheel. In particular, if a 
and a' are constant and equal, we shall have merely a movement of transla- 
tion, while if a' = 0, q; = constant we shall describe two concentric circles. 
If = 0, q; = 90°, the machine will describe a circle about the rear wheel. 
If, with these values oi a, a' the length / be made variable, and a given func- 
tion of s, the front wheel will describe a curve and the rear wheel its evo-^ 
lute. Thus a curve may be mechanically constructed from its intrinsic 
equation. This principle is used in certain integrating machines, and the 
author has invented an integrator for drawing trajectories involving this 
principle, which will be described later if he succeeds in having it built. 
In the case of the automobile we have 
a' = 0, p = /etna, ds = IctnadO, 
and if as before w^e assume the mode of steering to be a = as we may 
integrate and obtain the intrinsic equation of the curve, 
Idd = tan (05)^5, lad = log sec as. 
Developing the tangent in series we may see how much this differs from 
the Comu spiral. The curve has been drawn graphically from the differ- 
ential equation, but since on a sixty-foot chord it differs but slightly from 
