MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 523 
Also, r,” = CB? = CR? + RB? = (CQ + PB) + (AP — AQ)? 
= (P, + Be)” + (Ge — %)? 
= py + 2p, By + Py” + 72” — Py” —~29%% +7, — By 
ry = 7 + 7 + 2p, Py — 2% % 
Since the first curve is fixed to the paper, we may find the angle 6, 
Thus é, = DCB = DCA + ACQ + RCB 
RB 
=o + tant + tan 5 
é, = 6, + tan — + tan7! sr 
Thus we have found three independent equations, which, together with the 
equations of the curves, make up six equations, of which each may be deduced 
from the others. There is an equation connecting the radii of curvature of the 
three curves which is sometimes of use. 
The angle through which the rolled curve revolves during the description of 
the element d s,, is equal to the angle of contact of the fixed curve and the rolling 
curve, or to the sum of their curvatures, 
as, + d 8» 
R, 


il 
But the radius of the rolled curve has revolved in the opposite direction 
through an angle equal to d 6,, therefore the angle between two successive posi- 
, : d s: < : 
tions of 7, is equal to —d%, Now this angle is the angle between two suc- 
2 
cessive positions of the normal to the traced curve, therefore, if O be the centre 
of curvature of the traced curve, it is the angle which d s, or d s, subtends at O. 
Let OA=T, then 


ds, r, d 6, ds. ds, as 
s OE Phy i ae aia, 
RA eee 7 Aas tacit 
ab, 1 i 1 dé 
7? —_2 = + — 7 eal 
d 8, Ja R, R, 4 So 
Ge eae i 
PANGS Gare Sheen? Ytare 
As an example of the use of this equation, we may examine a property of 
the logarithmic spiral. 
In this curve, p = mr, and R = -, therefore if the rolled curve be the 
logarithmic spiral 
VOL. XVI. PART V. OF 
