524 MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 



(iat 5 ail nan 
sae NA roe aa R, si Fe 
it She 
ao) Re 
‘ AO 
therefore AO in the figure = m R,, and R7™ 
1 
Let the locus of O, or the evolute of the traced curve LYBH, be the curve 
OZY, and let the evolute of the fixed curve KZAS be FEZ, and let us consider 
FEZ as the fixed curve, and OZY as the traced curve. 
Then in the triangles BPA, AOF, we have OAF = PBA, and a == — 
therefore the triangles are similar, and FOA = APB = 2 therefore OF is perpen- 
dicular to OA, the tangent to the curve OZY, therefore OF is the radius of the 
curve which when rolled on FEZ traces OZY, and the angle which the curve makes 
with this radius is OFA = PAB = sin™ m, which is constant, therefore the curve, 
which, when rolled on FEZ, traces OZY, is the logarithmic spiral. Thus we have 
proved the following proposition: ‘“ The involute of the curve traced by the pole 
of a logarithmic spiral which rolls upon any curve, is the curve traced by the 
pole of the same logarithmic spiral when rolled on the involute of the primary 
curve.” 
It follows from this, that if we roll on any curve a curve having the property 
p, = m,7,, and roll another curve having p, = m, 7, on the curve traced, and so 
on, it is immaterial in what order we roll these curves. Thus, if we roll a loga- 
rithmic spiral, in which p = mr, on the nth involute of a circle whose radius is a, 
the curve traced is the n + 1th involute of a circle whose radius is / 1 — m’. 
Or, if we roll successively m logarithmic spirals, the resulting curve is the 
n + mth involute of a circle, whose radius is 

aJl1—m? J/1—m,? J ete. 
We now proceed to the cases in which the solution of the problem may be 
simplified. This simplification is generally effected by the consideration that the 
radius vector of the rolled curve is the normal drawn from the traced curve to the 
fixed curve. 
In the case in which the curve is rolled on a straight line, the perpendicular 
on the tangent of the rolled curve is the distance of the tracing point from the 
straight line ; therefore, if the traced curve be defined by an equation in «, and y,. 
Us SF aero ond a BG 2) 

