528 MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 
d 
«, for r,, and multiply by «,, equate the result to = , and integrate. 
Thus, if the equation of the rolled curve be 
—>Ar”+4 ete. + Kr?24+Lr + Mlogr + Nr + ete. + Zr” 
=n Ar — ete 2K — Lr +MrouNiete tn Zr 
r 
oY =n Ae — etc. 2K 2? —- Let + M+Natete +nZer 
x 
t= Ag sete, +2Kat—Llog#++ Me+4N a + ete. + BS ee 
a n+ 
which is the equation of the fixed curve. 
; : Cy. 
If the equation of the fixed curve be given, find 7. in terms of x, substitute 
ry for x, and divide by v, equate the result to ae and integrate. 
Thus, if the fixed curve be the orthogonal tractory of the straight line, whose 
equation is 
this is the equation to the orthogonal tractory of a circle whose diameter is equal 
to the constant tangent of the fixed curve, and its constant tangent equal to half 
that of the fixed curve. 
This property of the tractory of the circle may be proved geometrically, thus— 
Let P be the centre of a circle whose radius is PD, and let CD be a line constantly 
equal to the radius. Let BCP be the curve described by the point C when the 
point D is moved along the circumference of the circle, then if tangents equal 
to CD be drawn to the curve, their extremities will be in the circle. Let ACH 
be the curve on which BCP rolls, and let OPE be the straight line traced by the 
pole, let CDE be the common tangent, let it cut the circle in D, and the straight 
line in E. 


