930) MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 
of the other curves is equal to a, and the angles which they make with the radius 
at the point of contact are equal, 


d 6, d 4, : 
n= a @ Serj) ands, 5 oe moe 
i 
as, eee if 
am a dr, . 
: dé,. 
If we know the equation between 0, and 7,, we may find es in terms of 7,, 
1 
= T») 
substitute + (4+ 7,) for 7,, multiply by , and integrate. 
Thus, if the equation between @, and 7, be 
7, = asec 4, 
which is the polar equation of a straight line touching the traced circle whose 
equation is 7” = @, 
then 




Ct ee a 
a7 ie (7) a) 72 +2 7, 
a 


Now, since the rolling curve is a straight line, and the tracing point is not in 
its direction, we may apply to this example the observations which have been made 
upon tractories. 
Let, therefore, the curve * = 2; be denoted by A, its involute by B, 
and the circle traced by C, then e is ae tractory of C; therefore the involute 
2 : : 5 = ee 
of the curve r = == is the tractory of the circle, the equation of which is 
slave 2 5 : 2a 
0 = COs? | ef “—1. The curve whose equation is 7 = z_y Seems to be 
among spirals what the catenary is among curves whose equations are between 
rectangular co-ordinates ; for, if we represent the vertical direction by the radius 

