MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 531 
vector, the tangent of the angle which the curve makes with this line is proportional 
to the length of the curve reckoned from the origin; the point at the distance a from 
a straight line rolled on this curve generates a circle, and when rolled on the cate- 
nary produces a straight line; the involute of this curve is the tractory of the circle, 
and that of the catenary is the tractory of the straight line, and the tractory of the 
circle rolled on that of the straight line traces the straight line ; if this curve is 
rolled on the catenary, it produces the straight line touching the catenary at its 
vertex ; the method of drawing tangents is the same as in the catenary, namely, 
by describing a circle whose radius is a on the production of the radius vector, and 
drawing a tangent to the circle from the given point. 
In the next case, the rolled curve is the same as the fixed curve. It is 
evident that the traced curve will be similar to the locus of the intersection of 
the tangent with the perpendicular from the pole; the magnitude, however, of 
the traced curve will be double that of the other curve; therefore, if we call 
r, = p, 8, the equation to the fixed curve, 7, = ¢, 6, that of the traced curve, we 
have, 
r, = 2p, = 6 — cos 10 = by — 7 + sino 
Op 2 To 
also, 41 = 7% 
Lf th 
1 0 
2 2 
Similarly, r, = 2p,=2r,2¢= 47% ay, (7), # = 6 — 2 cosh 
: 79 " "9 "0 
Similarly, (Bee 7) ie aa etc. = 27, a 
1) vo 
and Pn — Po 
vr %o 
—1 Po 
0 
-1 Pn 
§, = 8 — m COS 


Now, cos" is the complement of the angle at which the curve cuts the radius 
VOL. XVI. PART V. 6x 
