522 MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 







the letters s, s. s; to denote the length of the curve from the pole, p: p, p; for the 
perpendiculars from the pole on the tangent, and q 9 9; for the intercepted part 
of the tangent. : 
Between these quantities, we have the following equations :— 






= Sri 
r= a +H Stan y 
x =r cos é = ¢ sin 6 
Meo t e af 
= 2 ele. dé = LP l 
c= Ves Gi) = fu @ 
bape is Ao _ _yda—ady 
oe) oe Ny P= (dx)? + dy? 
dé 
rar 
dé be adxt+ydy 
sei SY 1 V@a+ Gy 
2 d 2\ 3 
nx (22)')3 (a+ (22)')5 
dé R dx 
R= 2 Fp a @ y 
ey yeae eee Py 
re has de dx 
We come now to consider the three equations of rolling which are involved 
in the enunciation. Since the second curve rolls upon the first wethout slipping, 
the length of the fixed curve at the point of contact is the measure of the length 
of the rolled curve, therefore we have the following equation to connect the fixed 
curve and the rolled curve,— 
8, =, 
Now, by combining this equation with the two equations 
= (%) = y 
(HERS pon BEE GO) 
it is evident that from any of the four quantities 0, 7, 6, 7, or %, y, &, y,, We can 
obtain the other three, therefore we may consider hese Gun fies as known fune- | 
tions of each other. ? 
Since the curve rolls on the fixed curve, they must have a common tangent. — 
Let PA be this tangent, draw BP, CQ perpendicular to PA, produce CQ, and 
draw BR perpendicular to it, then we have CA=r, , BA=r,, and CB=r,; CQ=p,, 
PB=p,, and BN=p,; AQ=q,, AP=9, and CN =q,. 
